Shooting a dog with buckshot from a reasonable distance is no worse than spanking a kid in which you probably have a problem with as well. Buckshot from a reasonable distance will not penetrate the dog it only stings. Shooting at a running dog with ammo capable of killing will at least half of the time result in a crippled dog or a wound that won't heal without a vet. You don't know "shotgun ammo" because some of the most deadly forms of projectiles can come from a shotgun. Slugs and Double 00 to name a couple. I am attempting to save the dog by stinging him with birdshot buckshot in 7 1/2 or 8 loads from a reasonable distance. You have to know the gun and load and its capability. Don't shoot when its too close and only shoot when its running away. Common sense that a gun owner should use. Your suggestion to shoot to kill will result in at best a dead dog or mortally wounded one that will hopefully die sooner rather than later. Now tell me about your choice of gun and the proper load to dispatch the dog keeping in mind you will be firing a gun in a residential setting.
I have not had a problem with any domestic animals that have harassed my chickens. I do not let them free range, we put thought and time into the large yard they are in and the yard is inside my horse pasture a ways away from the barn, so the horses coming and going scares much of the populace of wildlife away. However, the horses at night aren't in the barn, they are on the other side of the pasture, so we kept that in mind.
What we have shot, cleanly killing and no 2nd shot was needed, is a fox. I used a Ruger rifle, with 10/22 sub sonic ammo, a scope, and a silencer. Please Okallthis4eggs, keep in mind, I live in a rural setting, my closest neighbor is a half acre to an acre away on 2 sides of my farm. I am an FFL, so gun safety and those around me are first and fore most. My husband was an avid hunter till work, bills and a family became priority. I have been taught, as our children as well, if your going to take the safety off of any weapon, be prepared to use it and use it well and shoot to kill. If it is an animal, you shoot it and kill it, you don't try to "sting him", or anything else as it has been shown in this post and other posts, that the animal will and often does, come back.
If your in a residential setting, put up a fence and keep your neighbors dogs out.