If one person were to take something from a chest would another person find an empty chest or will there be something different on the other players end?

As far as I know, they're moving more towards the old-school mechanic where you'll have to wait for items to respawn in containers. As for the timing, no idea, but if harvestables are any indication, it will be fairly fast, but it may not be in the exact location it was the first time around.

This is a great question and one that I have been wondering about as well . . .

I think maybe the chest will disappear and reappear somewhere else to keep randomness and unpredictability of loot locations. I hope that some chests will be cleverly hidden in logs, crags, foliage, and other places; I'm sure they will. Perhaps set quests will have respawn spots and times for certain chests.