If the people handing out traffic tickets are not police officers of some sort, then they won't have any legal right to conduct an arrest and detention. Or at a minimum they would be conducting a citizen's arrest. Few, and maybe no jurisdictions currently allow a citizen to detain another citizen forcefully over a traffic violation.
If Berkeley is just starting up another police force, whose sole job will be to make traffic stops, then that police force would of course be able to do what the current police do. And in effect they would have a version of CHP. Maybe they will call it BSP for the Berkeley Street Patrol.
Berkeley's plan seems half baked. That isn't surprising considering where it was hatched.