How does passive insight/perception skills work? I've read about it and asked people but I keep getting different answers. Any help?
Note: I'm going to just talk about perception here. Insight works exactly the same way, except that NPCs will roll bluff to try to trick the PCs, rather than stealth to sneak up on / past them.
To calculate a player's passive perception, first calculate their perception modifier: half their level (rounded down) plus their wisdom modifier plus 5 (if they're trained) plus any bonuses from their race/theme/background/items. You should get a number somewhere between -1 and +35. The player's perception rating is 10 plus this number you just calculated.
Now when something is trying to sneak up on the player, it rolls stealth. If it rolls higher than the player's passive perception, then it successfully hid from the player. If it rolls less than or equal to the player's passive perception, the player notices it.
Note that Q&A subforum of the official D&D forums has a stickied thread explaining everything you ever wanted to know about stealth in 4e.