The Fed doesn't really operate in secret. You can get the data. It reports its obligations, assets, and operating profits each and every year. Its profits, which the Fed earns solely from the banks, are then turned over the Treasury to help fund the federal government.
And no matter how public the company, nobody will every fully understand the minutia of a corporation. The same for the Fed. The Fed chair doesn't know what Barney Smith in KC does at his desk all day.
Until an outside, independent auditor certifies the books of any public company, the reports on operations should be viewed with skepticism.