I was a contract inspector for the State of Florida for 8 years but that was a special deal. (only state projects). During the go go years Lee County Fl was contracting out some of their inspections. Nobody really made me an offer I couldn't refuse although a couple of the IAEI officers worked for the engineering company that was doing it and "bumped" me a couple times.
The real rules were that they would own the whole permit from start to end but the reality was they would just get inspection cards now and then for things that started with Lee County. That was when a muni inspector might walk out in the morning with 40 cards spread out over half the county.