It is a false dichotomy, that assumes owners of an organization are required.
There are several kinds of orphan entities, such as trusts, foundations and more.
In the USA at the national level, only Congress is able to incorporate businesses, and they do this in one-off charters, Act by Act. So it was incorporated by a public body - Congress - while the wording of its charter leaves it very autonomous and orphaned. It is not owned by the public, it is not owned by the private sector. Simultaneously, Congress created another public body called the Board of Governors of the Federal Reserve, which is owned by the public, and interfaces between the autonomous orphan entity and the public. It is the one with the appointed Chair, who occasionally reports to Congressional committees.
Congress can amend the charter of both organizations at any time. They don't and it is a line they do not cross. The alternative is the politicisation of monetary policy, which is a heavy distraction for Congress, far more than fiscal policy. It didn't work well before the autonomous central banks and other better alternatives haven't been presented.
To further complicate things - in the minds of those perturbed by the Federal Reserve's role in this country - the Act allows for collaboration with private banks as shareholders, with a 6% dividend. Yes, banks have been earning 6% dividends from their shares of the Federal Reserve for a century. These shares do not convey voting rights. This was to encourage participation in the Federal Reserve system, and any new system would need to be extremely competitive and enticing to encourage banks to participate in that instead. For context, think about America in 1913 when the Act was passed. Banks existed and had their own payment networks all around the country, and the Federal Government wasn't in the business of this at all. The idea of inherent fealty to whatever the US Government represented simply did not exist, the idea of an omnipotent US Government didn't exist. Impressionable children were not taught this in schools and bankers then and now obtain better benefits from not thinking this way. Instead, America was a burgeoning society, that recently got bailed out by JP Morgan himself, now trying to get into finance. It had better be very convincing to the banks!
Like any orphan entity like a foundation or trust, there are people that control it together, in accordance and restrained by the charter. The Federal Reserve is a system, controlled by regional directors who are selected/elected. In each region:
Three directors are selected by the Board of Governors of the Federal Reserve System to represent the public. These directors must reside and conduct business or other activities in the District. They represent the interests of labor, consumers, commerce, manufacturing or agriculture. They may not own stock or serve as a director of financial institutions.
Six directors are elected to the board by the Bank's shareholders, which are the member banks in the District. Of those six directors, three are representatives of the District's banks, and three represent the public (like those selected by the Board of Governors). The three elected public directors may not serve as a director, officer or employee of a financial institution.
Yes, the System is able to purchase certain kinds of securities (or whatever Congress allows, such as the new amendments for direct money to citizens as in the stimulus packages) and whoever it buys from now has newly created money that is diluting the money supply. The System is also able to trade the securities it has purchased for existing money. It is interested in not causing rampant inflation, but this is an inherent possibility, but it is fortunate that it has inherited a larger economy than the rest of the world, and there are people willing to accept its dollars and it just selling them into liquidity keeping the dollar's relative purchasing power amongst other currencies steady.