Private set supporting adding elements and proving elements exist
If we put in the set $H(x||r)$ instead of x where $r$ is a large enough random value (w.l.o.g assume each x is in the same length). Then, each player can add values without revealing it and the same item can be inserted by different players without knowing it (a collision might happen with negligible probability).
Then with access of the third player, one can prove that the value is indeed in that group.