How are coordinate systems built physically? How is one to imagine a physical realization of a space time coordinate system? Is there some efficient way to think about this?
If I just consider the 1+1 one case, I would imagine that there is a rigid meter stick and on each interval of the stick there is a clock attached, so each clock has a unique position label, further there would be some means of storing and measuring data at each clock.
I could also consider infinitely many observers carrying a clock with them and they would have some means of knowing their positions from some common origin, maybe by exchanging light signals with each other, keeping track of the time light takes to bounce back? 
Basically, does it always boil to assuming that there is some means always available to setup up a coordinate system and how it is done does not matter much. 
 A: "There has to be an assumption in the thought experiment that measurements would be made using some sort of network of observers."
It is exactly how it should be done. In many cases observer in special relativity is not a physical person, who has exact spatial localization. The observer is the whole reference frame, which consists from a number of observers (it is often said - a team of observers). Each of these observers takes certain position and possesses a clock. Then these observers synchronize these clocks by beam of light. The beam of light travels from observer to observer and they adjust their clocks accordingly, since they know velocity of light c and distance the beam travels, so they can calculate how much time it took for the light to go from clock to clock.
This observer is often called as a rest frame. The observer (or rather team of observers) conducts measurements in his own rest frame.
Very often people think that observer "sees" dilation of moving clock, for example a clock from distant position sends readings to observer, he calculates distance, considers velocity of light  e.t.c. Or they see passing by light clock in which light travels by hypotenuse. You can see many reflections like that, even from advanced researchers. It is wrong.
To measure dilation of moving clock an Observer (or team of observers) must have at least two clock in his frame, let's say clock C1 and C2. When moving clock passes by clock C1, he compares readings in immediate vicinity. When clock passes by clock C2, he compares readings again. If moving clock and C1 showed 12 PM at meeting, moving clock will show 3 PM and C2 shows 6 PM when they meet. This is how time dilation works. SINGLE moving clock dilates relatively to a set of synchronized and spatially separated clocks, not vice versa. Set of clock runs faster from the point of view of single clock.
This article emphasizes this important detail at page 6 (6)
http://isites.harvard.edu/fs/docs/icb.topic455971.files/l09.pdf
Notion of Observer in SR
https://en.wikipedia.org/wiki/Observer_(special_relativity)
Animation:
https://en.wikipedia.org/wiki/Time_dilation#/media/File:Time_dilation02.gif
Related articles:
https://arxiv.org/ftp/physics/papers/0512/0512013.pdf
Good article that gives correct explanation:
http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%20Physics/Chapter039.htm
"Two spatially separated clocks, A and B, record a greater time interval between two events than the proper time recorded by a single clock that moves from A to B and is present at both events."
To be exact, single person can conduct measurement of time dilation and length contraction without hiring assistants. The Observer can measure dilation of moving clock if that clock emits radiation just by looking at it at right angle to direction of motion of the source (at points of closest approach) and to measure deviance of frequency - so called Transverse Doppler Effect.
Single person can also measure length  contraction of relatively moving rod by means of taking a picture of this rod at the moment, when the rod is at point of closest approach. He has to attach two flashes on the ends of the rod and launch these flashes simultaneously either in his frame or in the rod's frame.
There are some other details, but that's another story.
