I have a very similar thing happening now at my residence in Grayson County.
Next door was vacant for a couple of years and then last year a family moved in. They brought with them one young female Pit Bull. I was not too concerned since their property was fenced.
At that time there was a male Pit Bull that roamed the neighborhood and had already killed another smaller pet dog. So a problem actually already existed.
Well, the first day the new occupants arrived, I find their dog in my yard. I go over and tell the guy that the dog cannot roam the neighborhood. It will be shot by someone (maybe me)? He said that there was a small hole in his fence and the dog got out that way. He would fix the fence.
Time went on and the next thing was that the feamle was in heat and you guessed it, the male came around and did his thing. Right out there in the street in front of my house.
Present day.......now there are 3 pups about 10 months old next door and they routinely get out of the fence and into my yard with the Bitch.
If I wanted dog poop in my yard, I would buy my own dog poop making machine. I had bought a paintball marker with the idea that I may have to use it some day to scare them away and best to use that rather than wasting ammo. I came home from a long trip away on the 5th and got into the house with no problems. But, when I went out to the car to get something, as I came back to the house the three dogs were in my back yard barking up a storm and slowly advancing on me. I was not armed!
I made it into the house and grabbed my new tippman98 custom paintball marker and came back out and opened up on them. I had not sighted it in and missed most shots, but did manage to hit one right in the butt as they all took off. I have since sighted it in with the attached red dot sight and 14" sniper barrel and I am ready for the next time.
So, now I don't know what I am going to do. Continue to use the paintball marker and run them off as necessary, or bring out the 870 and do some damage. The property is in the country and there is no animal control to call. The Sheriff's office has known about the original Male Pit Bull running loose and has told the neighbors to just kill it next time it is out and nothing will be done.
Suggestions?
Further to the original poster's thoughts.....
I personally would never use a pistol to put down a dog. Only if it was a last resort and I had no other way out. Reason is because a dog is fast and hard to hit with a pistol. If you are actually intent on killing a dog, a shotgun is your best choice with 00 buck.