We let people have guns so they can execute people with no judge, no jury, no witnesses, and no questions asked so long as they can convince someone they felt threatened or thought they saw a gun.

No, we let people have arms (which includes guns, knives etc) so they can defend themselves. If we have an inalienable right to life then we also have a right to defend our lives. If we don't have the right to defend our lives then we have no right to life.

yet almost no effort at all is made to do the same to prevent private executions.

You are incorrect. If a person meets the criteria for self-defense then it is not a private execution but self defense. If a person does not meet the criteria for self-defense then they are charged with a crime.

http://dictionary.law.com/Default.aspx?selected=1909
https://www.lectlaw.com/def/d030.htm
http://criminal.findlaw.com/criminal-law-basics/self-defense-overview.html