Actually Pits were very popular at the turn of the century. I can't remember the specifics but one served as unofficial mascot with some unit in WWI and wound up being decorated. He toured in parades after the war. They were used for their strength of character in war posters. But their popularity faded sometime after WWII and when they returned to public notice it was because of unresearched media attention(none of us are familar with that, eh?).
The media has a bad tendency to portray any dog attack at this point in time as being by a "Pit". Unidentifiable Heinz? Pit. Short haired? Pit. Etc and so forth, because it grabs the attention of a public who are, to put it mildly, not generally too bright where news is concerned.
Anyway, the thread title would have been better had it not listed any specific dog since, if it's gnawing your ankle off, it doesn't really matter what you call it. As for shooting "a dog", if you're in city limits you're screwed unless it's literally self defense. If not then generally speaking you're OK. Do you shoot on your property now? Doesn't matter what you're shooting at, then.
As for the dog itself, if the owner wanted it they'd keep it under control. Shoot, shovel and shut up. It's just beyond me why people seem to think they need to involve the police in this sort of thing? We've been shooting dogs, and cats, all my life and long before that. My neighbor's dog is scheduled to die this weekend, when it again returns to eat more of my chickens. I told him twice, which is more than I'd do for most people and certainly more than I would expect from anyone if my dogs were off killing and attacking.