What is a "Private" Airport?

[MidlifeFlyer]

Is there a listing at FAA.gov or from another source?

You can go to the FAA's Airport Data page and generate a list of the areas you want. The dataset you get will have fields for "Ownership" and "Use" with either PU (Public) or PR (Private) populating them.

Whether the airport is public or private use is, of course, also on the sectional chart.