On what specific astronomical observations is the Fermi paradox based? This is a question about the Fermi paradox, a topic that can attract a lot of opinions. However, the scope of this question is strictly limited to a specific factual topic, and I've edited it quite substantially to try and make this clearer.
Enrico Fermi originally introduced his paradox in the following terms:


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*We expect life / advanced civilisations to be common in the Universe

*If technologically advanced intelligent life existed, we would expect it to visit Earth

*Therefore its absence requires an explanation. (There is a huge body of literature that attempts to do this.)


However, it seems common in modern popularisations to replace (2) above with:
    2$*$. If technologically advanced intelligent life existed, we would be able to detect it remotely.
My question is about this point (2*). The claim here is that we should expect extraterrestrial civilisations to have observational signatures that we would have already detected if they were present. I am somewhat skeptical about this claim, and so I would like to know more details about the assumptions and measurements behind it.
The reason I'm skeptical is that, as far as I'm aware, SETI has concentrated for most of its existence on detecting deliberate signals, i.e. radio signals that have been directionally beamed at Earth with the objective of making contact with us. However, one can think of all kinds of reasons why signals of that kind might not currently exist, even if advanced civilisations are common in the galaxy.
If we rule out this possibility, what else is left? Listening to popular accounts of the Fermi paradox, I'm left with the impression that the presence of extraterrestrial civilisations would just be obvious if they were there. But until the 1990s we could not detect exoplanets at all, and even now we can obtain very little information about their composition or surface details. It does not seem at all trivial to make measurements that would reveal the existence or absence of an extraterrestrial civilisation.
Therefore my question is, which specific observations - by which instruments - have been cited to support the idea that extraterrestrial civilisations do not exist, or are not common, in our galaxy? Note that I'm asking for observations that we have made already, rather than ones we potentially could make in the future.
Of particular interest to me are the observational signatures of megastructures such as Dyson spheres, or of waste heat from civilisations sufficiently large for that to be detectable. It seems to me that, given some assumptions about the size of the civilisation involved, these might have observational signatures that would allow them to be ruled out to some extent, and I am interested in whether this is possible and/or has actually been done. But more broadly, I am interested in published, referenced versions of claim (2*), so that I can understand the assumptions and the astronomical measurements behind them.
 A: SETI was always based on very long odds, as well as being poorly funded because of dismissals like those in some of the comments here. The main signal listening is done in a narrow band that has least attenuation by hydrogen atoms and dust, on the assumption interstellar signals would be sent there. The uncertainties are very high. It has been more about reasoning our way through what we might expect, than really realistically looking. 
The paradox is really that, given the size even of just our galaxy, the gigantic number of stars, we would expect some clues. Techno-signals could include laser or radio noise or other light or gamma ray bursts, nuclear or antimatter explosions, space mirrors and star-occlusion not linked to an appropriate planetary mass, unstable elements in atmospheres. Our expectations of numbers are arranged into the Drake Equation, and in general recent observations have increased the already high expected numbers of civilisations it outputs.  
Life started earlier than previously thought on Earth. Only a few hundred million years after the end of cosmic bombardment, suggesting that with physical characteristics of our planet (goldilocks zone, rocky with molten iron core, plenty of water) life is quite likely to occur. And happened over a very short time compared to the age of the universe. Whether life occured on Mars or Europa will add information. 
It has turned out most stars have planets, and that rocky planets are common, as we have got better and better at observing solar orbital wobbles and planetary transits. Still looking at 'super earths' or bigger though.  Some information can be gleaned about atmospheres too, and none with oxygen have turned up yet. First exoplanet with magnetic field observed 2014.
I must say I think the questions you ask are exactly the right way to go about looking at the issue. The Fermi Paradox is up there with Dark Energy and Dark Matter as one of the great outstanding mysteries. Narrowing the relevant unknowns, and focusing attention on research that will narrow them further, shouldn't be treated as fringe science, but as addressing some of the key issues of cosmology and biology. 
Some success with atmospheres, in 2014 the first magnetic field inferred en.m.wikipedia.org/wiki/HD_209458_b and in 2017, super-Earth GJ 1132b https://phys.org/news/2017-04-atmosphere-super-earth.html
Breakthrough Initiative, Watch project aims to look at 'nearby' stars for biosignitures inc oxygen en.m.wikipedia.org/wiki/Breakthrough_Initiatives Antimatter makes gamma rays, hot topic for astronomers, closely watched. Very high energy bursts remain unexplained, and interstellar comms is a candidate. 
Our own current level, sure [we wouldn't have observed]. But, we expect a higher level of technology to persist, for a long time, with increasingly noisy signals. For that to not happen for other occurences of life millions of times in our view, we then have to assume there is a en.m.wikipedia.org/wiki/Great_Filter, or a much better way of communicating/travel/fighting based on physics we don't know yet. Or, we are in a simulation.
Edited to add: This consideration of whether the Simulation Hypothesis is the best answer to the Fermi Paradox has some great quantitative statistical work on expectations for various scenarios of intelligent life occurence and distribution https://www.stat.berkeley.edu/~aldous/Research/Ugrad/mark_yu_haihan.pdf
A: As others has already suggested, it is not an easy task to detect civilization that is not communicating. A prerequisite for it on interstellar distances would be its astrophysically  significant footprint such as a Dyson sphere or some other signature.
A relatively recent search for Dyson spheres:


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*Richard A. Carrigan Jr., IRAS-based whole-sky upper limit on Dyson spheres. The Astrophysical Journal 698.2 (2009): 2075, doi, arXiv.


From the abstract:

Searches have been conducted for both pure   (fully   cloaked)   and   partial Dyson   Spheres   in   the   blackbody temperature  region  $100 \le T \le 600\, {}^\circ\text{K}$.  When  other  stellar  signatures  that  resemble a Dyson Sphere are used to eliminate sources that mimic Dyson 
  Spheres very few candidates remain and even these are ambiguous.

Sixteen “candidates“ were identified. The largest one Sun bolometric distance, $D_\text{sol}$ for those sixteen sources is 118 pc. Potentially this search could have found a Dyson sphere with luminosity of our Sun within the distance of 300 pc.
Another recent survey based on WISE data and investigating galactic scale (Kardashed type III)  potential waste heat emissions, reported in the series of papers:


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*Wright, J. T., Mullan, B., Sigurdsson, S., & Povich, M. S. (2014). The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. I. Background and justification. The Astrophysical Journal, 792(1), 26, doi, arXiv.

*Wright, J. T., Griffith, R. L., Sigurdsson, S., Povich, M. S., & Mullan, B. (2014). The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. II. Framework, strategy, and first result. The Astrophysical Journal, 792(1), 27, doi, arXiv.

*Griffith, R. L., Wright, J. T., Maldonado, J., Povich, M. S., Sigurđsson, S., & Mullan, B. (2015). The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. III. The reddest extended sources in WISE. The Astrophysical Journal Supplement Series, 217(2), 25, doi, arXiv.

*Wright, J. T., Cartier, K. M., Zhao, M., Jontof-Hutter, D., & Ford, E. B. (2015). The Ĝ search for extraterrestrial civilizations with large energy supplies. IV. The signatures and information content of transiting megastructures. The Astrophysical Journal, 816(1), 17, doi, arXiv.
The “Ĝ“ (G-HAT) in the title is an “acronym“ for Glimpsing Heat from Alien Technologies.
From the abstract of the second part:

We present a zeroth order null result of our search based on the WISE all-sky catalog: we show, for the first time, that Kardashev Type III civilizations (as Kardashev originally defined them) are very rare in the local universe. More sophisticated searches can extend our methodology to smaller waste heat luminosities, and potentially entirely rule out (or
  detect) both Kardashev Type III civilizations and new physics that allows for unlimited “free” energy generation.

As a result a catalogue of 93 candidate galaxies has been identified. Followup 


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*Garrett, M. A. (2015). Application of the mid-IR radio correlation to the Ĝ sample and the search for advanced extraterrestrial civilisations. Astronomy & Astrophysics, 581, L5, doi.


demonstrates that a majority of these sources are associated with galaxies in which natural astrophysical processes are dominant, so that the conclusion is

that Kardashev Type III civilisations are either very rare or do not exist in the local Universe.

A: I'll limit this "answer" to just address the part about SETI. Basically, we would not have detected our current level of civilisation even if it existed around Proxima Centauri, apart from under very fortuitous circumstances. Therefore SETI has very little to contribute in terms of the Fermi paradox.
A "blind" search for radio signatures that relies on detecting random radio "chatter" and accidental signals generated by our civilisation is hopeless at the levels emitted from Earth.
The SETI Phoenix project was the most advanced search for radio signals from other intelligent life. Quoting from Cullers et al. (2000): "Typical signals, as opposed to our strongest signals, fall below the detection threshold of most surveys, even if the signal were to originate from the nearest star". Quoting from Tarter (2001): "At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars)...". 
So the answer to your first question is a less-than-useful 1 light year.
We do emit stronger beamed signals in certain well-defined directions, for example to conduct metrology in the solar system using radar. Such signals have been calculated to be observable over a thousand light years or more. But these signals are brief, beamed into an extremely narrow solid angle and unlikely to be repeated. You would have to be very lucky to be observing in the right direction at the right time if you were performing targeted searches for such signals from other civilizations.
Hence to detect signals from even the nearest star systems you would have to scale up the "radio-leakage" from the Earth by an order of magnitude or more.  new radio telescope projects and technology like the Square Kilometre Array may be capable of serendipitously detecting radio "chatter" out to distances of 50 pc ($\sim 150$ light years) - see Loeb & Zaldarriaga (2007). This array, due to begin full operation some time after 2025 could also monitor a multitude of directions at once for beamed signals. A good overview of what might be possible in the near future is given by Tarter et al. (2009).
Addendum 
Wright et al. (2018) have done a quantitative estimate of what fraction of the "haystack" that all SETI programmes have searched adequately for needles. They consider distances out to 10 kpc and conclude that only 1 part in $10^{17}$ of the multi-dimensional haystack volume has been probed so far, assuming that transmissions could come from anywhere.
The paper contains many arguable assumptions and scenarios, but the basic point is that the lack of "success" of SETI cannot be used to support a Fermi paradox-type argument 
