I would suggest you learn about the US Federal Reserve System. I will agree the with you that the Regional Reserve Banks in the US are privately owned. They are actually owned by banks, about 40% of the banks in the US collectively own the 13 or 14 Regional Reserve Banks. However; owning shares in a Regional Reserve Bank is not like owning a shares in, let's say Apple. Regional Reserve Bank shares cannot be transferred, assigned or sold. Also, the entire Fed Reserve System is controlled by a Board of Governors controlled by the US Government.
Now let's talk profits. Each of the banks that own the Regional Reserve Banks put up their money. In fact; to own shares in a Regional Reserve Bank, a shareholder needs to put up something like 4% of its own combined capital. And for that, the shareholder is entitled to a 6% dividend. For example, if Goldman Sachs total combined capital is $100 billion, Goldman would put up $4 billion of its own capital to get shares in the Regional Reserve Bank. And that entitles them to a dividend of $240 million. Yes that sounds like a large dividend, but they put in $4 billion to start. Profits of the Reserve Banks then go to the federal government. In 2015 the Reserve Banks made $100 billion in profit, about $3 billion was paid out in dividends and $97 billion was transferred to the Federal Government.
So basically; private enterprises (i.e. banks) put up the money to get the Reserve System up and running, the Government controls the Reserve System, and the Government gets a very big chunk of the profits generated by the Reserve System. Basically, it's like you putting up all the money for me to start a business, you own all the shares, you cannot sell the shares, I control the business, you get 6% return on your investment and I keep all the profits. Sweet deal for me!
And the Canadian central bank system is almost identical to the US. We just structured it differently in that we do not at the outset require private investment into our reserve system. We used the granting of Schedule 1 licenses to make the large banks comply with the Government's requests when it comes to moving liquidity in and out of the system.