I've read this a dozen times now and still don't understand what it's trying to accomplish... you end your turn, don't draw a hand, then draw what your left-hand opponent plays (assuming similar deck composition). And then on your next turn, anything that draws cards is useless, so hopefully you're not trying to play an engine. And then I guess you play a turn with their cards, then after you buy something you take another turn?
Like Possession, you're getting an extra turn that's approximately as good as one your opponent played. Exchanging a normal turn for a Smugglersy turn would be a Ruins power level card, so it needs to be an extra turn.
It's actually fine for your opponent to play an engine, because searching for your deck actually doubles the effect of cards that draw. It's a little counterintuitive until you count it out. In the design without this problem fixed: I have 4 coppers in hand and play a Smithy, draw 3 silvers, and play all of those cards to buy a Castle, and you watched all of that with Covetous, on your turn, your hand will begin with 4 Coppers, 3 Silvers, and a Smithy. Should you play the Smithy and draw any treasure, you'd be doing better than the player you're coveting.
This behavior is retained for cards like Farming Village and Envoy. I think that is ok, because players are not forced to pick up Farming Village or Envoy, and it adds some nuance to the decision. They're not forced to buy cards with +cards either, but the design space is so loaded with it they might have to play Big Money if they don't want to allow the doubling.
If I have two of these in play, it would appear that I get the effect twice, so if the LHO plays a Silver, I can dig two Silvers into my hand?
This is impossible, similar to Tactician. You can't even do it by discarding some kind of unreleased Tunnel that draws you cards, since the moratorium on drawing cards would already be in effect.
Even Mission or Outpost + Hireling can't do it..
Other than the starvation rule, what stops them turning the game into a stalemate where they don't play anything, so you don't draw anything until you buy a Copper?
I'm kind of not sure exactly how that would play out. I'm pretty sure it is suboptimal play to -consistently- decline the Heirloom's potential, gaining a 5$ and having your opponent gain a 4$ should be a better deal than handing out a Copper most of the time. Governor feels a lot more exciting than a 5$ card that says "+1 Action, eachother player gains a Copper", right?
But I'm thinking the requirement that you buy a Copper should be relaxed probably. To tell you the truth I tried to set things up against this card in some ways because I know how much people hate Possession, the idea that the Possession player is playing more of a stables deck and less of a Chapel deck was part of my attempts to sate bloodlust.
Also, this would be extremely vulnerable to any top-of-deck attacks. "What's that, you don't have a discard pile any longer? Sucks to be you."
You still draw Rabble, Spy, or Scrying pool itself, but mostly, I considered this and figured the number of top of deck attacks is acceptably low enough. Maybe it's not. It at least seems nonobvious to me. Some of the top of deck attackers cause this card to transform itself into the top of deck attacker that harmed it, due to "when you play" timing. For instance, if you get Bandited, you can go get a Bandit, and gaining a Gold and trashing a Silver is a respectable outcome from your nonterminal 5$ workshop.