change the controls between two characters on the scene
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This would depend on a lot of factors - what engine you are using, how you set up the controls, etc. Regardless, you'll need to have some kind of "switching" mechanism - an action the player can take to perform the switch (examples: press the space bar, click a button on the HUD, maneuver the character to perform an in game action, etc). Whatever this "switch" is should kick off an event that perform the control switch. One easy way to do this would involve swapping a Boolean flag (eg: isPlayerOneMoving = not isPlayerOneMoving), then use that Boolean to control what each controller inputs does.
Example:
OnSpaceBar_Pressed():
if (isPlayerOneMoving):
PlayerOne.jump()
else:
PlayerTwo.jump()