The party makes progress together ... kind of... Each character has their own quest log and you can actually get quests on one character that do not show up on others even on single player. (Theres a left and right button on the quest log window above portraits) The caveat to that of course is plot triggers those seem to be global.
This can also get confusing because if you talk to a quest NPC one one character you can progress/bug it for another. For example the grieving mother quest you can mix up the dialogue by talking to her with separate characters missing the trigger you got from another NPC conversation that would allow you to near instant complete it
This also leaves room for exploits as I was able to do the free potion quest on all 4 of my characters on SP. I could also repeat failed influence checks by switching characters.
