Fiction

Police didn't even exist in the form we know them when the first mysteries were written by Edgar Allen Poe and Wilkie Collins. Almost all of the "famous" detectives of mysteries are not police--Poe's Dupin, Doyle's Sherlock, Christie's Poirot & Marple, Stout's Nero Wolfe, and so on.

If anything, amateur sleuths dominate mysteries compared to professional investigators (cops, PIs, lawyers). Every career field imaginable has yielded an amateur sleuth--chefs, librarians, bookstore owners, bounty hunters, doctors, nurses, teachers, reporters, postal workers, engineers, archaeologists, historians, scientists, beauticians, DJs, government employees, writers themselves, art dealers, clergy, and etc.