What is the difference between a private and public exchange?

A for-profit company can be either private or public. In the case of the for profit private company, the exchange is owned by shareholders who usually have a stake in the operation of the exchange - i.e., the business that the shareholders operate depend on using the facilities of the exchange, such as investment dealers.

You or Iwould not own any shares in the for profit private company. A public company issues shares to the public and anyone can own shares in the company, including you and Ias well as the stakeholders who need to use the facilities of the exchange to carry out their business operations.