So, in general, playing Possession amounts to getting an extra turn. Obviously there are crazy exceptions with Island, Masquerade, Ambassador, and Trash-For-Benefits. But anyway, how different would Possession be if it simply gave you an extra turn instead of giving your opponent an extra turn that may as well be your turn?
Of course, you need a stipulation, similar to Outpost, to prevent infinite turns. Since playing your opponent's Possession on his possessed turn doesn't work (like you want it to!), we can simply word it like this: "If you are not currently taking an extra turn from Possession, take an extra turn after this one."
So, it seems like it would be stronger in some ways; and weaker in some:
Why it's stronger:
-Your opponent can't stop you by messing up his deck.
-You get to earn VP tokens.
-Playing attack cards in your extra turn still hurts your opponent.
-You cycle through your deck faster.
-You can play Durations without helping your opponent.
Why it's weaker:
-No Masquerade/Ambassador/Island tricks.
-No Governor / Vault / Bishop / Council Room / Oracle / Spy tricks (where you can improve your own deck on your opponent's possessed turn).
-Trash for benefit cards aren't as great; you can't just freely trash a Colony for +11 cards without losing points.
-You can't screw with your opponent's next turn via Herbalist, Scheme, Alchemist, etc.
-You have to have a deck with real buying power; you can't build a strategy around just paying multiple Possessions per turn.
What other things would be different? My thought it that as a whole; the own-deck version is stronger.