Our fence borders 4 neighbors (we are on a cul-de-sac), and we'd take full responsibility for any portion of it that needed to be replace. We built the fence in conjunction with one neighbor to save on costs, and if that part were damaged, we'd certainly confer with them before any adjustments, etc.
Any other portion would be our responsibility in my opinion. In fact, we pulled our fence in 18 inches on one side to avoid a drainage swail - we cut the grass behind there so the neighbor on that side doesn't have to maintain what is technically our responsibility.
Our HOA has strict guidelines for types of fencing, height, styles, so we're not really at liberty to make such decisions. But, I'd never ask or assume that a shared neighbor would have to pay to replace it. Honestly, I believe it should be the responsibility of the person who hit it (the affected portion, at least).
People can really be strange.