The fastest way to get rid of a nieghbor's pit bull is to file a claim after a bite. Most insurers will demand the dog is removed from the home after such a claim.
What I found odd is that after we had moved into the neighborhood we noticed that while everybody got along, there appeared to be a dividing line between two distinct groups. The pit bull couple were in one group, and the previous owners of our house were in another. We later discovered that the pit bull had mauled the previous owner's dog, and there had been a pretty nasty lawsuit that went to trial. It was brought out that this was the second dog that she had mauled.
So when I was bitten and went to a clinic for treatment, the folks at the clinic said I needed to tell them the dog's owners' information and they had to report it to Animal Control. The AC folks came out a few days later and interviewed the neighbors, but the dog is still here. We thought certainly the dog would be put down.