If you are under a persistent damage effect, you (typically) get a flat DC15 roll to remove it. Various things can trigger this roll and/or change the DC, but it for sure comes right after you take the persistent damage.
Damage from multiple sources stacks only if it is of a different type. That is, I could have 2 effects of persistent fire damage 2, and one of acid 1d6, and I would take 1d6 acid, and only 2 fire.
Does it take a separate roll to remove each instance of persistent damage? Or can it all be removed with one successful check?
What if they are of a different type? If I have two instances of persistent fire damage and one of acid, is that one, two, or three rolls to get me out of the hurt locker?