Whose yard did the fight occur in? If it was the neighbor's yard, then you are liable since your dog was off leash and off your property. The fact that the neighbor's dog was off leash would be irrelevant since you are not required to keep a dog on leash on its own property.
As for not breaking up the fight - I understand the inclination to try and break up a fight (I've done it myself), but it's actually a pretty foolish thing to do. Your boyfriend's medical insurance may ask for contribution from your neighbor - whether or not they will I can't say.
Who did the neighbor report the fight to? If it was animal control or the police, then you will have to wait to hear from them - depending on what your neighbor told them, they may not even contact you except to make sure your dog is up to date on shots. If it was reported to your homeowner's association, I have no idea what they might do about it.
Dog fights happen - there is no "public disgrace". Make sure you can prove your dog is appropriately trained and put up a fence.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
"while trying to break it up, my boyfriend ended up being bit by the neighbors dog. "
Which means that the injury should have been treated and documented. It would be assertable either as a direct claim against the neighbor, or as a counter-claim in a suit brought by the neighbor.
The foregoing is for general information purposes and does not establish an attorney-client relationship.