The Fed is a hybrid system, both public and private.
The Fed is owned by its member banks. They get to elect board members to the regional boards. Profits accrue the banks, kind of. The Fed doesn't pay a dividend so those profits never get monetized. So de facto profits go the Treasury.
However the Board of Governors is appointed by the president. This limits the amount of control the banks have over the institution. So what is ownership without access to profits or control?
A analogy would be the local co-op grocery store. The more you buy the more of the store you own. However this comes back as store rebates and the insiders still have control.