Unless I am mistaken (I could be thinking about a different game) but when in a party GW2 used to play the cinematic for both players when one player started the conversation - has this changed? Is it a toggleable option or am I just imagining the entire thing?
The cutscene plays for every member of a party, regardless of previous completion if it is a quest cutscene. For the cutscene to be skipped all players must skip it, if this is not the case the players who skipped will be able to walk around while the others finish the cutscene.
Those who have not skipped are denoted by a film reel above their head.
There are cinematic that cannot be skipped and you need to watch them. Then you have cinematic that you can skip them.
When you are on the same area (map) with your friend and you are on the same Story quest on of you can be leader of the story and start the quest. Then a pop up message will come to your party-friend to accept if he likes to do the story with you.
Then Story start normally and after it finished it requires from the non-leader (the guy who started the story) if he approves to continue the story or he likes to stay on this one (so he can do it again)