Unity RoE Quests

Is it the intent of the quests to only be able to be completed by the quest holder?
For example, if attempting to complete either Vanquish Monsters X or Vanquish "Family Type" or Vanquish "Family Type" with "Magic/Ph." Damage, it seems to only register the credit if you make the killing blow.

My friend and I both have the quest active (Kill Beasts with Magic Damage), and yet it onyl registers the kills for him when he finishes it off with a spell. If I have Shantotto out for Trusts and she kills it, only I get the kill and not my friend. Why is this?

It's not the quest holder. The person landing the killing blow has to be in the same Unity as you.