Instructions
Verify that the achievement in question does not say "Single Player only," as these "LEGO Batman" achievements cannot be gained as player two. For player two to get these achievements, you would need to have both players sign out and switch positions before signing back in.
Verify that player two has not already achieved the "LEGO Batman" achievement in question, as achievements can be gained only once. You can check this by pressing the Xbox 360 "Guide Button" on the controller, then press "left" once, and select "Achievements."
Ensure that player two is signed in using a Gamertag as opposed to being signed into "LEGO Batman" as a "guest." If no specific Gamertag is used, no achievements can be gained. To sign in, press the Xbox 360 "Guide Button" on the controller, press "X," then select the appropriate Gamertag and press the "A" button.
Ensure that player two has played the entire level, as opposed to "jumping in" in the middle of the level. Some achievements in "LEGO Batman" require that both players participate in the entire level, or at least require the player to be in the game prior to the activity related to the achievement.
Restart the level in question with both players staying in the game for the duration of the level, or at least to the point at which the achievement would be awarded.