Normally, once a game is started, it is not affected by any other game slot. However, a new game started in a new player slot will have the settings from the last player slot that was active. If you start a game on hard, and then change it to easy, you will have a game that still has some characteristics of a hard game.

The only way to truly start a new game with different settings than the other player slots is to change the settings in the new game to what you want and then click Start Over on the menu. That way, you know that everything will be set up properly with those settings.