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I have limited experience with 4e but 30 years DMing, so my answers above are based on that, and upon a dash of common sense. I know that PCs and monsters to not make passive checks, as that is what "passive" means. The score is a DC that someone using an active skill has to beat in order not to be casually spotted (for Stealth) or caught in deception (Bluff), for example
I think for 4th ed, yes the creature or player who is stealthing makes the check against the target's passive perception check.
From the rules, it appears that you make that check at the end of the stealthed creature or player's turn. This would seem to be in addition to the check you make when, for example, becoming stealthed in the first place.
From "Targetting what you can't see", PHB pg 281:
Invisible Creature Uses Stealth: At the end of a concealed creature’s turn, it makes a Stealth check opposed by your passive Perception check.
If you beat it, you know there’s a creature present that you can’t see, and you know the direction to its location.
If you beat it by 10 or more, you know exactly what square the creature ended its turn in.
The concealed creature also makes a Stealth check if it takes an immediate action or an opportunity action.