Objection 1. It would seem lawful for a private individual to kill a man who has sinned. For nothing unlawful is commanded in the Divine law. Yet, on account of the sin of the molten calf, Moses commanded (Exodus 32:27): "Let every man kill his brother, and friend, and neighbor." Therefore it is lawful for private individuals to kill a sinner.
Objection 2. Further, as stated above (2, ad 3), man, on account of sin, is compared to the beasts. Now it is lawful for any private individual to kill a wild beast, especially if it be harmful. Therefore for the same reason, it is lawful for any private individual to kill a man who has sinned.
Objection 3. Further, a man, though a private individual, deserves praise for doing what is useful for the common good. Now the slaying of evildoers is useful for the common good, as stated above (Article 2). Therefore it is deserving of praise if even private individuals kill evil-doers.
On the contrary, Augustine says (De Civ. Dei i) [Can. Quicumque percutit, caus. xxiii, qu. 8: "A man who, without exercising public authority, kills an evil-doer, shall be judged guilty of murder, and all the more, since he has dared to usurp a power which God has not given him."
I answer that, As stated above (Article 2), it is lawful to kill an evildoer in so far as it is directed to the welfare of the whole community, so that it belongs to him alone who has charge of the community's welfare. Thus it belongs to a physician to cut off a decayed limb, when he has been entrusted with the care of the health of the whole body. Now the care of the common good is entrusted to persons of rank having public authority: wherefore they alone, and not private individuals, can lawfully put evildoers to death.
Reply to Objection 1. The person by whose authority a thing is done really does the thing as Dionysius declares (Coel. Hier. iii). Hence according to Augustine (De Civ. Dei i, 21), "He slays not who owes his service to one who commands him, even as a sword is merely the instrument to him that wields it." Wherefore those who, at the Lord's command, slew their neighbors and friends, would seem not to have done this themselves, but rather He by whose authority they acted thus: just as a soldier slays the foe by the authority of his sovereign, and the executioner slays the robber by the authority of the judge.
Reply to Objection 2. A beast is by nature distinct from man, wherefore in the case of a wild beast there is no need for an authority to kill it; whereas, in the case of domestic animals, such authority is required, not for their sake, but on account of the owner's loss. On the other hand a man who has sinned is not by nature distinct from good men; hence a public authority is requisite in order to condemn him to death for the common good.
Reply to Objection 3. It is lawful for any private individual to do anything for the common good, provided it harm nobody: but if it be harmful to some other, it cannot be done, except by virtue of the judgment of the person to whom it pertains to decide what is to be taken from the parts for the welfare of the whole.
Source: Summa Theologica (New Advent)