It is highly unlikely to accomplish this (though technically possible in theory).
Computers generally only focus on one window or application at a time. The window that is active is the only one that receives input from the keyboard/mouse, and (almost always) the controller. This is why you can play a game in one monitor with a text editor open in the other monitor and not have to worry typing garbled text there. The computer would only send input keys to one application at a time: the one with focus.
There are two ways to circumvent this, which may or may not be options (most likely not):
- The game in question is specifically designed to continue running and receive input after losing focus: Most games are specifically designed the exact opposite of this. Usually games are designed to pause when it loses focus.
- You have a third party application that governs input methods and routes them to the appropriate window: I'm not aware of such an application, but I anticipate it could potentially exist out there somewhere.
Given the complexity of the configuration at hand, it would likely be easier to either obtain a second computer, or take turns using the one you have.