Pausing the game does stop everything in the game; however you have to keep a few things in mind about how pausing works in LDW games.

Each player slot must be paused individually. Pausing one game does not pause them all.

Leaving the game on the menu screen (or any other screen) does not pause the game, even though it seems like nothing is going on.

Going to the Options screen and pausing the game will completely pause the game, but you must click Done on the Options dialog and (to be extra safe) quit the game.

If you intend to leave the game running, the safest and most reliable way to pause it is to press the space bar and verify that you see the message "GAME PAUSED" in the blue player comm area. However, any time you leave any application running, it can be affected by computer crashes, etc., so it's best to pause the game and then quit until you're ready to return to it.