Constables can attack criminals freely without increasing their infamy.
A criminal always has a bounty on his head, from the government in any case, sometimes in addition from another player. If a bounty was placed the Constable will receive a reward from the person who issued the Bounty once the Criminal is captured (killed). Constables lose their status if they kill too many people and gain an infamy of 2000 or higher. They may not apply to be a constable again until they have returned to normal (infamy below 2000).