Kitchen Finks

If a permanent has multiple instances of persist, they'll each trigger separately, but the redundant instances will have no effect. If one instance returns the card to the battlefield, the next to resolve will do nothing.

If multiple creatures with persist are put into the graveyard at the same time (due to combat damage or a spell that destroys all creatures, for example), the active player (the player whose turn it is) puts all of their persist triggers on the stack in any order, then each other player in turn order does the same. The last trigger put on the stack is the first one that resolves. That means that in a two-player game, the nonactive player's persist creatures will return to the battlefield first, then the active player's persist creatures do the same. The creatures return to the battlefield one at a time.

The persist ability triggers when the permanent is put into a graveyard. Its last known information (that is, how the creature last existed on the battlefield) is used to determine whether it had a -1/-1 counter on it.