I don't know if we are required to call the sheriff's dept. when we shoot a dog or not, but I figure it is better to call them and let them know so they can come out and see the situation and that you killing the dog was justified, better that than the dog owner calling the cops with a my neighbor just shot my poor pet doggy scenario because then the cops will show up at your house with it already in their head that you are a animal abusing criminal and will be trying to justify arresting you. Also if you plan to try to get compensation for your killed birds and damaged coops/fences it helps to have a police report and testimony from an officer.
