As the colonies were not yet unified under a central government, each colony created its own money and its own public central bank.
This is explained in the following quotation.
"The American colonies were an experiment in utopia in an uncharted territory, you could design new systems and make new rules. Paper money was already in use in England but it had fallen into the hands of private bankers, who were using it for private profit at the expense of the people. In the American vision of this new medium of exchange paper money was issued in lent by provincial governments, and the proceeds were used for the benefits of the people. The colonist's new paper money finance, a period of prosperity that was considered remarkable for isolated colonies lacking their own silver and gold. By 1750. Benjamin Franklin was able to write off New England, and there was abundance in the colonies in peace was raining on every border. It was difficult and even impossible to find a happier, more prosperous nation on all the surface of the globe."