Whew, good question. It is frequently the local state that does that determination.
We have a lot on an airport, a Privately owned, Public use, airport. The actual airport property is a condominium, owned by the condominium association, in that all of the owners own an undivided share of the runway, and taxiway. Keeps someone from getting a divorce and putting up a fence on the runway. So, we are one of the owners. The airport manager, however, is not one of the owners. He was established by the state, many years ago, and continues in that role today. We, the owners, have talked about replacing him, with someone from ownership. It is a simple, yet complicated process, it involves taking a test, passing the test, then petioning the state to change the designated airport manager. If the state agrees to the petition, then the change takes place. Note, however, that a considerable effort has to take place, and there has to be a good reason to replace the individual. Since the airport manager has a house off the runway, and is present in the area, and hasn't done anything "wrong", we haven't expended the effort to replace him.
If the airport is privately owned, the airport owner could be the manager. However, if that person was never designated by the state (and it varies from locale to locale), it could very well be that the state, and hence the feds, have designated some outside individual or organization as the "official" airport manager.
Oh, if it is privately owned, but not public use, then federal or state funds will be harder to get and use. In other words, if the VFR map says "P", don't look for the feds to do anything. If it is public use, no "P", the funding is available. Just red tape to get it.