Same as any public versus private object, like a road. A private road is owned by an individual or a corporation, a public road is owned by the town, city, village or other public entity. Exactly the same concept applies to a septic system, utility, building etc.
Rather than a public sewage system where the waste water is treated and returned to a body of water like a river, it all empties into a septic system which, I imagine works similar to a private system but on a larger scale. There are probably some design differences due to the larger amounts of waste and water.