OP yes, you should expect your neighbor's to pay your vet bills. About 4 years ago our cat was killed by our neighbor's pit, our cat was in our yard under our tree, their dog was out with no leash and NO training, he saw our cat and ran into our yard, chased her across the street into another neighbor's yard and where she tried to climb the tree, unfortunately the dog was faster and he caught her. He literally crushed her to death. We rushed her to the emergency clinic but she died on the way, when we got there they took her back and the vet came out and asked if she had been run over by a car, I said "no, it was our neighbor's pit bull" and she said "I'm calling animal control, you need to do the same. I'm sorry for your loss, she was a beautiful kitty" and she was. We did call animal control and report the incident, we did go to a dangerous dog hearing and the dog was deemed dangerous. The owners were allowed to bring him home but they had to have a $1 million dollar home owners policy (which they couldn't get when the ins. companies found out why they needed it), they had to build an enclosure to keep him fenced at all times and they had to do a bunch of other things. The dog came home for about 2 weeks and then he was gone. I don't know what they did with him, I think they gave him to someone else.

In the time period between the dog killing my cat and the hearing the dog was released to their custody but he was able to jump the 6 foot fence to get out, he chased me and DBF many times, it got to where we didn't let the girls go outside at all until we made sure that dog was in their yard and then we hustled to the car to leave. He was scary!!!!! He was NOT a nice dog. I was so relieved when he was gone.
Now they have another one and this one is not nice either. At least this one stays fenced tho.
OP I did not ask them to pay the bill I was hit with for our visit to the emergency vet (yes, even tho she was gone when we got there I still had to pay them), all I asked was for them to get rid of the dog or have him put down.....I was so worried about my girls getting attacked by that dog. I used the bill as a bargaining chip, told them I would forget the bill if they would get rid of the dog. Now I don't know if they did that because that's what I asked but I do know they only had him for a couple of weeks before he was gone.
I have 2 cats and 2 dogs, if my dogs bit someone or attacked someone's pet I would absolutely
offer to pay the bills.....