Depends on the state or municipality. There are some places where the duty weapon is not personally owned but kept at the employer's or client's location. In those situations they issue it when you report for duty. Look into the Heller case circumstances for instance to see how state and local laws have an impact on this situation. It is a mess.
As someone mentioned earlier, insurance companies are heavily involved in thisprocess due to liability issues. This is one reason that so many companies prefer hiring off-duty officers or even arranging their "security services" through anoffice of the PD itself. In many instances the PD even handles billing and scheduling of staff, includingsupervisors.