Right now, Eddie and his boys run the place. They've got the guns and the dynamite, so they call the shots.
Meyers is found in the NCR Correctional Facility. He claims he is the former sheriff of a town to the west. The Courier may rescue Meyers from the NCRCF which can result in Meyers becoming the new sheriff of Primm.
Meyers was the sheriff of a town in the wastes before being arrested by the NCR for "taking justice into his own hands". He is now serving his sentence at the Correctional Facility, not involving himself with the Powder Gangers who took over the jail.
If you tell Meyers that Primm needs a sheriff and agree to his conditions, he'll leave the Correctional Facility and wait for you to return with news of his pardon. If you then choose a different sheriff, Meyers will become a permanent resident of Primm with the same dialog as when he was inside the NCRCF. In this case he'll mostly hang out with Ex-Deputy Beagle behind the Vikki and Vance Casino.
If Meyers is the sheriff of Primm in most of the endings, the final cutscene tells that he was a good sheriff, but sometimes someone died without concrete evidence. If you completed Veni, Vidi, Vici, he will fight off the Legion until the end, but Primm is quickly overrun.
A cut speech exists with in the game's files, meant to be delivered by Meyers to a small group of inmates close to Meyers should you recruit him to be the Primm's new Sheriff. In it, Meyers calls upon the other inmates to join him in forming a new police force in Primm, calling them "honorable and looking for redemption" despite not having "experience with law enforcement... at least not on the enforcement side."