The game has been updated to fix issues regarding playing co-op on a keyboard or when playing with a controller+keyboard combo. An important note is that "player 1" and "player 2" use different keys on the keyboard regardless of what setup you use. They are as following:
Player 1
Player 2

When playing single player, the controls for Player 1 is used, with the addition of using Q to change character.
If the game does not update for you, you can restart Steam to force it.
Update March 14
To be clear - you can now play the game coop using the following setups:
If you own a controller that does not work with the game, please let us know.