Since it's a dog, it's mostly a question of destruction of someone else's property, and cruelty to animals statutes. There would be a question of civil liability, but I doubt any court would seriously entertain a claim by someone who's dog was attacking people with lots of witnesses.

On the criminal side, some cruelty to animals laws have exceptions for protecting yourself or your property from injury but these can vary. For example in some places it may not be legal to shoot a dog that's only attacking your dog and not people. Some other laws say it's only a crime to "recklessly" kill someone's animal, with no specific exceptions for defense of people or property, it would be up to a court to find that you were not reckless.

There can be discharge of firearms ordinances in cities as well, depending on how those are written they may not cover the situation of shooting an animal that is a threat.

Anyway so it strongly depends on where you are, but in a lot of places you could be charged with something if your prosecutor wanted to pursue it, or sued in small claims court. Most likely your actions would be defensible if it really was attacking people, and you'd win in court, but it's still possible to be caught up in a lot of headaches.