Here's the official word on it : https://www.sec.gov/answers/gopriv.htm

The part that is usually not explained is that there are state-by-state, and company-by-company, mechanisms to compel the left-over minority of shareholders who do not exchange their shares to be "squeezed-out" and paid the same cash value everyone else got. After this all of the shares are held by the private owner and the security is delisted/unregistered from the public markets