You can't really advise anyone on when it is or isn't legal to shoot a dog unless you have researched the local animal control ordinances where they live. In some areas it is legal to shoot a dog simply for being on your property, in some places they have to be killing livestock in some places it is harassing or killing livestock. I for one am not going to wait for them to kill or injure one of my animals before killing them, if they are on my property and chasing or acting like they are chasing they are getting shot. Besides that if nobody sees it but you all they authorities have to go off of is you word of what happened and they will have a hard time proving you did anything illegal, also if you just shoot them and don't tell anyone anything there will be nobody to accuse you of anything.If there's snow, can't you see the tracks of what did it and where it came from as well as where it went?
You can't legally shoot a dog ( and most wild animals) unless it's in the act of destroying your property