Actually, the correct answer is "it depends on what the browser gives them" -- some older browsers give full paths, some newer ones give "c:\fakepath" for example.
Easy way to see exactly what facebook uses would be to fire up a http debugger and watch the traffic. The filename bit in the header is what your browser sends.
In any case, facebook probably doesn't even look at the filename at all. They just grab the bytes -- probably after inspecting the header -- and stuff it into their rather insane image storage system.