How does private central banks?

I think you don't understand banking.

The terms you are using don't really apply.

All the Central banks are Controlled by Governments, through the various Treasury departments (European Central bank is controlled by Departments in Brussels) so the Federal Reserve although nominally a private bank is controlled by the US treasury, and subject to US Laws,passed by Congress.

Likewise the Bank of England, Bank of Japan, etc etc, Bank of England is called a Public bank but is actually owned by H.M. Government as it had been Nationalised in 1945, however like the Fed, BoJ, CEB, it is run autonomously by the Bank's Governors but subject to Laws and directly answerable to the Treasury.

There are no totally independent central banks, whether private (ie public shareholders) or Public (owned by the respective Governments)

The other answers don't understand Banks, banking legislation and 800 years of history and Laws since Magna Charta 1215.

Private Central Banks are controlled by International Bankers like the Rothschilds. While Public Central Banks are government owned.

private banks are owned by private individuals and public central banks are owned by the government