You are mixing terms.
First, the companies that you are referring to are corporations. There are several types of corporations: S corps, small corps, large corps, and limited liability corps.
The limited liability corporation is usually a combination of other business entities, that could include S corps, sole proprietership, a partnership, or a corporation. The purpose of the limited liability corporation (LLC) is to shield the investors from any liability that may arise as a result of the agreement to form a working relationship with other business entitites. Usually, those entities assume the risk of liability in return for the LLC to commit capital (money) to the endeavor.
Private and Public refers to the stock of the corporation. All corporations have stock. Each corporation must decide how many shares they will put up for sale. If those shares are owned by one person, or family, or even another corporation, but the sale of those shares are restricted to the members who own them then the corporation is held privately. The public sale of the stock does not have to occur on an exchange. If the corporation's documents allow for public sale than a shareholder can sell his stock at any time to anyone for a price that either the corporations board determines or at whatever price the market will bear.
Selling on the exchange is a totally different subject. Ther are several exchanges, NYSE, NASDAQ, CME, to name the big U.S. exchanges. There are also exchanges in almost every country around the world. It costs money to belong to those exchanges and the members are usually very large corporations.
Not to be confused with Indexes. An Index is a measuring stick of the exchanges. For example, the S&P 500, is an index that measures the price fluctuations between the 500 biggest companies in the country as selected by a firm called Standard's and Poor. The Dow is another exchange. The NASDAQ has its own index. The Russell, and on and on.
I just know this stuff.