YES, Only under the Tort claims act which has to do with negligence done to third parties, In other words if a board member runs over a kid with his car the US Governement can not be sued."Examining the organization and function of the Federal Reserve Banks, and applying the relevant factors, we conclude that the Reserve Banks are not federal instrumentalities for purposes of the FTCA, but are independent, privately-owned and locally controlled corporations." [Lewis vs. U.S., 680 F. 2d 1239, 1241]
The law is like that, your residence/jurisdiction under voting laws is entirely different than your residence/jurisdiction under taxation laws.
BUT HEY, Can you tell ME WHAT OTHER TOTALLY PRIVATE ORGANIZATIONS HAVE THIS REQUIREMENT
The Board is of Governors of the Federal Reserve is a 7-member panel who is appointed by the President of the United States and confirmed by the Senate (12 USCA §241).
Thus, all the key monetary policy decisions -- the ones that affect interest rates -- are made by a government agency whose members are selected by the President of the United States. The Fed may be privately owned, but it is controlled by the government.