I guess the wording should be:
(...)
At the start of the next player's turn,
pass this card to him, leaving it in play,
and +1 card +1 action for him.
It's hard to make a good wording. You could go with "move this to his play area" which is odd, or "he gains this, leaving it in play". I think passing seems like the right term for what you do here.
(Edit: Just what Captain Stupendous said...)
On the card itself:
The idea sounds interesting. But you have to change the effects so that it is not a card you will play in every situation. Otherwise it sounds like noone buys this because it'd be "common property".
Maybe you can give the left player a penalty (or at least, a condition) if he wants this card to go into his deck?
Then the Duration type wouldn't even be necessary:
Diplomat (Action) $5
+3 Cards
+1 Action
The player to your left may trash a Treasure from his hand.
If he does, pass this to him.
This way it goes into his hand, so th Diplomat would travel a lot faster then...