When questing in any public zone, if you have an objective to destroy a certain mob or object you don't have to destroy it personally, you only need to be nearby when it is destroyed.
It should be possible for ZOS to change the design, so that a player character which has accepted a sharable quest which requires the character to kill a specific "boss" must inflict damage to that boss in a fight, to satisfy that requirement if the boss dies as its outcome. However, I doubt that they will, whether the design should "be that way".When questing in any public zone, if you have an objective to destroy a certain mob or object you don't have to destroy it personally, you only need to be nearby when it is destroyed.
But that shouldn't be that way.
I love questing and it isn't cool, when someone other than me can finish my quest, just becaus I am "nearby". What if I just walk around a bit in a zone where I have a quest, which I don't play actually, and someone finishes it because all I de facto need to do is kill some mob? And not every player wants to "go in cave - click here - click there - finish quest" some of us just want to look around, to explore and take things slow.
Outside of being grouped, you do have to get a hit in in the boss. It is the rare thing where you get credit for being nearby.