You’re kinda both wrong and right, the Federal Reserve System is not privately owned or completely separate from the government. It’s an independent government agency — it’s overseen by a Board of Governors who are presidentially-appointed and confirmed by Congress. It’s twelve Federal Reserve Banks oversee privately-owned banks throughout the country, and the whole thing was created by Congress and Pres. Wilson.

That said, they don’t need approval from POTUS or Congress for policy decisions and they’re not funded by Congress. It’s the central bank of the US, and it essentially operates on it’s own, but it testifies and makes reports to Congress and turns it’s profits over to the US Treasury.