Detectability of interstellar messages Recently a debate started whether it is a good idea to send more messages into space in the hope of having alien civilizations receive them. There are some predecessors, most notably the 1974 Arecibo Message to globular cluster M13, which is an attempt to bridge 25 thousand ly with 1MW of power at 2380MHz and 10Hz frequency modulation (total message duration: 3 minutes).
I'm interested in what type of antenna would be needed to detect such a signal in a distance of 25 thousand ly (i.e. dish diameter). There are probably a lot of factors involved, so please state them if you provide a calculation.
There are also people stating that we already broadcast on a continuous basis since the time we invented radio communications and especially TV broadcasting. Because that kind of signal is much less directed and of lower power, what would be the distance after which our TV signals would vanish in the cosmic noise and become undetectable?
 A: This paper contains an important analysis of the different trade-off between bandwidth and energy efficiency. The interesting conclusion from that paper is that the most energy-efficient way to send and receive interstellar messages (over flat spacetime) that maximise the bit-rate requires making the bandwidth of transmission very large. In particular, this means that the traditional flavor of SETI that looks for suspicious stand-alone modulated frequency gaussians, might be too naive and restricting, and maybe be also a sub-optimal way of communication
There are other known tricks to enhance the bit-rate of communications at very low power. Using gravitational focal points of stars, it has been calculated and predicted by several authors that even a low-power cellphone could be detected at 10 light-years! So if Bracewell or Von Neumann probes are keeping up a galactic internet, gravitational focal points is where we want to place observers to sniff signals.
A: Some numbers come from a review paper by Cullers (2000), who discusses the SETI Phoenix project. There, it is claimed that the Arecibo dish is capable of detecting a narrow band, coherent signal of $f=10^{-27}$ W/m$^2$ given a 1000 second observation. Assuming that this is an isotropic signal, then the implied power at distance $d$ is $p=4\pi d^2 f$, which means that $p \simeq d^2$ MW.
So, it is clear that unless a 1MW signal is highly beamed it could not be detected by our current technology even from a nearby star. (Actually, this number is out of date, the receiver at Arecibo is somewhat more sensitive now, but I can't find any numbers).
Of course we do emit more beamed signals. The Arecibo radar transmits at 1MW, but its equivalent isotropically radiated power is 20 TW. In other words, the Arecibo dish could detect the directed signals it emits (and of course does, when performing solar system metrology) at distances of  about 5000 light years  although the radar does not normally send a signal for 1000s.
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 out 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.".
A recent survey using the Green Bank telescope was able to rule out continuous signals (between 1.1 and 1.9 GHz) at the level of 8 (beamed) Arecibo radars from a large sample of 104 Kepler planet hosts at distances of ~1000 light years.
The next generation of radio telescopes use "phased arrays" to monitor signals from many directions at once and can perform wide-angle surveys much more rapidly. The SETI project is now using the Allen Telescope Array. The claim is that over 10 years it can survey a million stars with sufficient sensitivity to detect the Arecibo radar out to distances of 1000 light years.
It has been suggested that new radio telescope projects and technology like LOFAR and the Square Kilometre Array may be capable (using a month or so of observing time) of serendipitously detecting radio "chatter" at a few hundred MHz out to distances of 10-1000 light years and over a fair fraction of the sky - see Loeb & Zaldarriaga (2007). The SKA 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).
EDIT: I realised I didn't fully address the question. The Arecibo dish can detect the beamed signal you talk about in  1000s at a distance of about 5000 light years. The dish has a diameter of 304m. So to detect a signal that comes from 5 times further away which will be 25 times weaker would naively require a 1.5km dish (assuming the noise levels remain the same).
A: I recall a tech presentation on YouTube that I cannot find to link to after an hour of searching. It was about a laser being built by the military, and he said that it's only an order of magnitude (or something like that) to scale up the technology to a something that would be an interplanetary searchlight, and could also signal over interstellar distances with a 8 meter mirror on each end.
The point is that the laser is pulsed, with a very high frequency and puts all the power in the pulses so is brighter than the average power. But, the reflection can look for that cadence and thus pick out the signal even against a strong background. I did find hits on this general topic going back a few decades.
So, a normal sized observatory could see beamed signals over 50 lightyears (if memory serves) if you know exactly what to look for. It is readable but not conspicuous and, combined with digital coding protocols, probably not discoverable without a lot of processing power and very fine spectral channels.
