If you used a 870 trigger in an 1100, it would NEVER hold the bolt open, although it theoretically would allow the gun to function and fire, you'd just have to rack the bolt back by hand, hold it back, drop the shell in, and release the bolt every single time.
If you used an 1100 trigger in a 870, you'd lose the slide lock function, making it likely or probable that the gun would fire out of battery.
So no, they are not interchangeable except MAYBE in the most dire or emergency circumstances (like a life or death situation).
The easiest way to tell the frames apart is lack of a hole for the carrier latch pin, lack of a hole for the carrier latch follower, and lack of cutouts for the carrier latch. Carriers can be swapped, and the action bar release cut is commonly made for 1100 release trigger unset button, so neither are 100% proof of which trigger is which.