Probably best to ungroup to do class stuff. If your friend goes before you and you go in his, that'll spoil it for you, and vice-versa. Just group up for the class phases when they seem difficult. If one finishes early, they wait before moving on.
You definitely can't do class ones as one, since you don't want your friend screwing with your class story. Other players are spectators, same as when you go in the class phase of another class. You could potentially do them as opposing karmas just to see how the class stories vary along alignments, if either of you tends to the other karma alignment.
My Trooper is Light, it's fun, very patriotic and lawful good. But others do Trooper as Dark and also have fun.
"Sir,we are beginning our search for these rumored 'underwater villages'.They will not stay hidden for long". -Battle Droid Commander OOM-9!