My main beefs with Possession are:
1. It is anisotropic. Like Smugglers, but on steroids, Possession varies wildly in its effectiveness. Possessing a strong engine (buying two provinces) or someone who has scaling TfB (Forge a Province) are vastly better than possessing a slog deck. This means that in 3er or 4er you can be completely shut out of any chance to win if one person makes a Possession bait deck (e.g. Nv/Bridge) that you can't touch. Worst case scenario is where you can only possess a slog deck while the guy across from you can possess a megaturn deck. I love that Dominion, generally doesn't care about who is next to you at the table ... Possession really changes that for the worse.
2. Possession is pretty swingy. 6P is normally far harder to get than 8 or 9 (and normally 11 once you get a plat or two), there are far fewer degenerate hands for 6P so you can't substitute 2 coppers for a potion like you might with silver. Likewise, if you aren't drawing deck every turn, then you have the luck of drawing 6P or 8 hands when you still want Possession and the flip late game when you really only want provinces. It is not uncommon for one player to buy the first 3 Possessions.
3. Possession leads to the most heavy analysis paralysis in the game. When you control both hands something that used to be an auto-play (like Ghost ship) now becomes a question of do I hurt my next real hand or help my current Possessed hand. Likewise, Possession makes for really long games when you get to playing it many times.
4. Possession is fiddly. Just about every game mechanic has something that reasonable people will misconstrue for Possession. Coin tokens - hey those aren't part of my hand you can't spend them. Bishop - hey I played the card I get the VP chips. Durations - hey I don't start with just 3 cards for my turn and I can still use Outpost for a second turn. I trashed your Fortress that means it goes into my hand. None of these are correct and you arguably might be able to puzzle them all out just from the plain text reading, but way too many people have trouble with getting it all correct.
5. Possession has some absolutely killer interactions with certain cards - Masq and Ambassador are insanely powerful and there is little you can do to stop them (e.g. I Masq over a Masq, Possess you, have you Masq a Province for a Masq and the have you Masq over your Possession). Another troublesome shot is using the set aside clause to give you opponent nothing but dead cards - draw their deck with yours (Gov, Council room) or by trashing their Cultists and then trashing all their cards using say Forge leaves them with nothing for next turn.
In general, Possession has all of the bad things of each of several cards all rolled into one. The anistropy of Smugglers, the swinginess of Tournament, the analysis paralysis of Torturer, the fiddliness Black Market, and the abuse potential Masq ... all in one card.
