The federal reserve is a public institution, no matter how many people want to write semi-provocative articles about its status in government. The confusion over the fed comes out of the fact that the fed is, technically, a private independent institution. How many truly private companies are allowed to set rates for whole countries, or have their leadership appointed directly by the President.
The Federal Reserve is often held to be a mix of public and private elements. But the Fed is 1. Owned by Private Banks, which hold stock. These banks collect interest on their stock for lending their reserves to the Fed. The US government does not own the Fed. 2. The US Treasury, which is part of the US government, prints Fed notes, and then sells them to the Fed. Public sells the private Fed the notes that get distributed through the banking system. The US Treasury also holds an account at the Fed, where we deposit our tax dollars. Again, the Fed is the Bank of the government, but not the government itself. 3. If the Treasury borrows money from the Fed, it pays interest on that loan. The Treasury is the public, paying a private institution, the Fed, for open market lending practices. 4. The Public element of the Fed, the Federal Reserve Board, takes part in some of the decision making process, but usually the major decisions get made by the privately elected FOMC. In other words, while the publicly elected board oversees the Fed's activities, much like the EPA would oversee a businesses' environmental impact, it does not in fact own or control the business itself. It merely helps regulate it.