Does that sound like a private firm?
Yes it does, since you're talking about shareholders. Shareholders make up a private firm. Citizens make up a public entity subject to regulatory oversight - which the Fed itself refuses to acknowledge, BTW.
And add for your numbers, they are utterly meaningless for lack of context or verifiability since the Fed refuses to be audited - and gets away with that refusal precisely because they are a private firm, and not a government agency.