I have no idea what you mean by "private executions"
I have a right to self defense whether it is with a gun, a stick, a 2x4, a rock, a metal pipe, my fist or anything else I can get my hands on. If you attack me and I strike you in the head with my fist and you have a brain injury and die. Did I execute you? or was I defending myself. The use or non use of a firearm has nothing to do with if it was self defense or not.