> -
there's no rule that a museum is supposed to be publicly owned. 
  I do not know about other states, but in New York
- yes, real
 incorporated museums are public, officially part of University of the
 State of New York, and must follow the Rules of the Regents. 
 It's illegal there to start up an independent museum?? 
 
 Without USNY accreditation, it's next to impossible for a museum in
 New York State to obtain tax exempt status as a museum.  It's similar
 to the accreditation requirement for public lending libraries.