This thing is a little odd to me. Obviously we don't know exactly what happend, but as a dog owner, I pretty much feel that we are accepting a fair amount of risk by letting our dogs run off-leash with lots of dogs, many that we don't know. If a dog was blood thirsty aggressive and the owner let it near another dog, I would definitely think they are responsible for any vet bills, but dogs run into each other constantly at off leash. All the time. If my dog hurt another dog (even in play they sometimes tear ears or get bad cuts), I would offer to pay or pay part of the bill, but when I got a dog I decided (consciously) that I was taking on any potential emergency vet visits.
What's really weird about this story is that they first posted it as a "run-in" injury (which sounds right with a broken leg) but have since changed it to "attack" or other vague terms in some reports. I also find it odd that they say they can't afford the vet bills but are offering a cash reward for info on the owners of the other dog.
Based on the various reports here's what I could imagine happening. First, both owners are breaking the law by having their dog's off leash at that time and it's pretty darn dark by then (even before DST) so the owners of the other dog may not have even known something happened. I've never seen a dog really injured by these collisions, but I've also never seen it happen to a dog with more delicate than average bones (I assume this is part of why whippet and greyhound owners tend to keep their dogs away from big dogs. Many I know have had multiple broken legs from mild injuries). They may not haven even known it was broken immediately and let the dog keep playing (dogs are tough). The owners of the other dog may know what happened but they may also have no clue the dog was injured and that there are now posters around the park looking for them.
I've seen some people saying that any dog that causes injury to another is the responsible one for any bills, but there are lots of shades of grey here. I've seen a dog start a fight but end up with the worse injuries. Is the dog (and owner) that was just defending itself really at fault?
It's a horrible situation for the owner this injured dog to be in, but I hope they aren't pinning all their plans on finding and getting the other owner to pay all of these bills. And hopefully they will consider pet insurance in the future.