Local Save Files in Co-Op on Borderlands 2

Solution 1:

Is it possible to save player2's character?

Yes, that is normally what happens. I play split screen and both characters get saved.

Each player chooses a storage device when starting a game. The character data will be loaded from and saved to that storage device.

If I started a game and picked the xbox360 hard drive, my character data on the flash drive wouldn't be loaded and the game would think I have no characters.