When I possess my left player, which player takes 2vp from Baths, I or she?
Taking 2vp is after her turn or in her turn?
Tentatively going with, the regular player gets the VP, not the Possession player. The usual "when x" logic says the turn is over; there's no "end of turn" phase.
"At the start of your turn" abilities happen in your Action phase (not before your turn starts), so going by that I should think that "at the end of your turn" abilities happen in your Clean-up phase. Like GendoIkari said, it also sounds like the intuitive reading of the card.
Save and Possession are the other two cards that have "end of turn" abilities. If "end of turn" happens after the turn, then if I Possess you and make you trash a Province, setting it aside, and that's the last turn of the game, that Province is never put in your discard pile. It was set aside at a point when it was no longer yours, so I don't know if it's yours at the end of the game then? I guess it is actually, since it was said aside by
you (being Possessed); compare with Inheritance.