I don't think so.
I just tried it myself with several different chars on two seperate accounts - atfer I've looted a chest with a char, I definitely wasn't able to see any more chests with that char.
I'm certain that it wasn't just bad luck. I've stood with two toons at the same time at a spawn point and tabbed between two windows so I
know that there was a trove there; still it was invisible for the char that already have looted one.
Joining a party didn't help either.
Edit: Note that both screenshots were taken at the exact same time.
http://i.imgur.com/a1OaeYD.png
http://i.imgur.com/mE3mQoS.png