I think your other chars just needs to be present in the map and in the same party as your mage to see presents drop. When I did last year, i used multiple mages in different channels and cced on my looters every now and then to check for presents on ground. All were in the same party.
Just to be very clear as it was difficult for me to get an answer for this specifically: The farming character can have the quest active & their specific box in their inventory. The other presents will still drop, I don't plan on farming another one but I assume presents will no longer drop after I open them.
My bishop did not have the quest but did all the killing. I had 4 characters in the map and same party with my bishop. Each had the quest for different horoscopes. Each character was able to see their own present only, not each others'.
