It’s not state-run.
The Federal Reserve is a private, independent institution authorized by Congress and jointly established by its member banks, which are controlled by a number of commercial banks, subject to congressional oversight, and must work within the economic policy framework established in the United States.
The Federal Reserve, the United States Federal Reserve System, consists of the Federal Reserve Board in Washington, D.C. and 12 regional Federal Reserve banks located in major cities across the country.
The Federal Reserve is mainly composed of members of the Federal Reserve Board and the chairman of the Federal Reserve Bank.