The wiki sort of implies that you do the damage (it simply directs, 'Do poison based damage...' or 'Do disease based damage...') but I'm like 90% sure that it doesn't have to be the quester him or herself who is the direct source of the damage. I could be wrong, it has been a while since I completed the quest, but I'm pretty sure anyone you bring along can do the poison or disease dmg you need.