Who you have actively selected and control will determine what stats are used during conversations/cutscenes, IE if you need to do a charisma or intimidation or insight check it uses that character's stats. However certain cutscenes will use your main character (the one you selected when you started the game) no matter what.

Also passive checks throughout the world will use all characters but they need to be near the spot to use them so sometimes if your character you're controlling walks past a hidden cache somewhere and fails a perception check you can move your other characters into that spot and they'll all try. You can even go to camp and grab someone else for more chances if you really have some blind characters in your party.