Do health wells, chests, and other interactive objects that allies use count towards the achievements?

Solution 1:

Both Health Wells and chests only count for the person that opened them.

I just joined a multiplayer game to test this out. I started at 37 chests and 5 health wells, and my friend had 78 chests and 10 health wells. He opened a chest and, after a delay of 3 or 4 minutes, his updated to 79, while mine stayed at 37. The next chest I opened, and after a delay mine incremented and his did not. We found a pair of health wells, and noticed the same behavior. After a few minute delay, the person who interacted with the object got credit while the other person did not.