Central banks are usually owned by the country they operate in. If the would start selling shares to the public and try to become a public company, that would be nice. Personally I would definitely invest in a central bank. Their funding comes from the government and their loans to commercial banks are usually backed by the government as well. It's a pretty solid business model with a lot of government guarantees. A great way to make money for the owners.
I really agree with you that Central Bank of a nation belong to that particular contract but from my understanding I understand that Central Bank of nation is not entitled to have shares or shares to the public because share is something of of company not of a government sector accentra bank when we talk of Central Bank it is a place we are a country or a nation general fund is stored , because any of the state branches so it's not something but someone can say a share