Yes, they do. Sherman County Ambulance Service has teams of three personnel who rotate one week on call. They are paid a modest stipend to be on call and are paid an hourly wage of $15 per hour while on call (whether that time includes responding to a call or simply waiting for one).
The on-call emergency personnel cover weekdays from 6 p.m. until the following morning at 4 a.m. and 24/7 on weekends. The on-call system was designed to allow personnel to go to their regular jobs during the weekdays. During the weekday, there are no personnel who are compensated to be available, but people do still respond.