Who the heck do you think was at the head of the throng executing the divine sentence but the aggrieved party? There was no "state" and "judiciary" when God gave this law.
"Think not that I will accuse you to the Father. There is one that accuseth you, Moses, in whom you trust. For if you did believe Moses, you would perhaps believe me also: for he wrote of me. But if you do not believe his writings, how will you believe my words?" (John 5.45-47)
I am a Christian, I follow the teaching of Jesus Christ. Where did HE say I can murder my husband if he is unfaithful?
I hope your question is answered. It can't be made any clearer