two player games but only use one common joystick
As in two player games where each player takes turns alternating between player one and player two? (one player at a time)
If that's the case then yes, you should be able to just re-map the controls in each game that you are referring to. Re-map player two joystick to match player ones.
But if the "other guy" decides to grab ahold of the joystick while you are playing then obviously he'll take control and throw a game off real fast.