I had the rivalry event come up in my last settlement phase, and two of my people fought, causing a gaping chest wound to one of them and adding the rivalry event to the next year in the timeline.
At first I thought it would be like the skull eater event
Spoiler (click to reveal)
which designates the nominated target of the murder event
but as far as I can tell from the text on the card it just says add the event and the next year you would draw it and pick at random again from the returning survivors.
I feel from the spirit of the event that it is supposed to be the same two people, since you can spend endeavor to dual and get the rivalry over with. However, as written it seems like the rivalry craze is just spreading around the settlement and everyone wants to fight each other.
Have other people run into this event? How did you end up treating it?