I cannot verify that for sure, but here in FL at least, I've never seen anything other than psychedelic mushrooms growing on shit. And the other ones (the inedible ones) look totally different - sometimes flat orange fungi, or completely white with white gills = BAD!
Animals with multiple-chambered stomachs (cows, horses) can eat grass and mushrooms (therefore eating the spores) without the spores being destroyed by stomach acids. So when they poop, the spores are still intact and surrounded by a good fertilizer (the poop) to grow in...thats why they tend to grow in cow shit, even though the cow shit is not mandatory.
Oh, and if you do this, be careful when removing the shit and the shrooms from the field. Watch out for Fish and Wildlife control people, make sure you flick the caps (to drop more spores) before you pick them, try to make a clean break on the stem (snip them, don't just rip them out of the ground), and let me know how it goes. I've thought of doing this a bunch of times, and it sounds like it would work, so I'd like to know how it works out for you.