Woah wait up! Damage to the UR dragon is shared online?
Yes. You also get better loot than you would offline. From my experience the Ur-Dragon tends to be a bit more aggressive online, as well.
So, it's possible for you to fight him at 4 1/2 life bars, come back an hour or two later, and he's down to one life bar or something because other people have been fighting him.
Note that people have been farming the shit out of him with assassin, though. They go at night with all of the assassin perks on, maxed out gouge, and no extra pawns to get the extra bonuses and gouge the shit out of him so he dies crazy fast.