Is the Federal Reserve public or private? Good question.
Answer: It's neither.
In short, the Fed is an independent entity within the government.
Explainer: From one perspective, the Fed looks like a public institution: Congress created it in 1913 to maintain the stability of the financial system; the president appoints, and the Senate confirms, the members of its Board of Governors; and it's not out to make a profit—after taking care of expenses, the Fed hands off its earnings to the Treasury. Furthermore, the details of its responsibilities are subject to congressional oversight. Still, the Fed is rightly classified as an independent central bank. Neither the executive branch nor the legislature gets a direct say in its decision-making, and it pays for its own operations (primarily by acquiring U.S. government securities on the open market).
Soure: US government & Slate.com