In the very beginning of the game, and during certain activities, only Mickey plays.
What platform are you using? PS3 or XBox?
I take it that you are not using a traditional Nintendo Wii, because, even as a single plalyer, I go from starting the game in single player, then transferring to coop, on a regular basis (just to move Oswald out of the way or get him to do something):
1) When you start, with one player, you are holding the Wii remote with the nunchuck plugged into it; and the first blue light on the bottom of the Wii remote will be lit (indicating player 1);
2) Connect a second nunchuck to a second Wii remote;
3) Click a button, like the "A" on the second Wii remote, and the blue lights at the bottom will blink, and stop, with the second blue light will be lit (indicating that remote belongs to player 2);
4) While the second remote is on, press the "2" button of that second remote, indicating that that second player wants to join, and the screen will split, usually with Oswald on the right and Mickey on the left;
5) Press the "2" button of the second remote again, and you go back to single player mode;
6) If you fail to move one of the characters for awhile, that remote will turn off;
7) I check the battery level in my remotes, even during play, by clicking on the "HOME" button, and looking at the lower-right corner. Press "Close" to return to the game.
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It may be different on other platforms, where it keeps track of who gets credit for what, instead of just putting all the goodies into the same basket.