It is possible to privatize all or part of bus service but the resulting private company would have to be paid [[out of public funds) to operate the service. This is already being done, all over the place. Many school districts have contracted out school bus service to First Student or others. DDOT has contracted out its paratransit service to private operators for many years.

Generally the profit in a private contract transit arrangement comes from imposing operational efficiencies of various types: better purchasing arrangements with parts suppliers, changes in contracts with various classes of employees, etc. So it's not inconceivable. I don't see it happening any time soon though.