In general, all securities offered in the U.S. must be registered with the SEC or must qualify for an exemption from the registration requirements. (Securities and Exchange Commission)
This seems to actually be a multi-part question. First: All stocks that trade on public centralized exchanges (other than over-the-counter stocks of small companies that trade between dealers and may be localized in trading locales) trade in all 50 states and, in fact, all over the world.
Second: All initial public offerings of stock for public sale on centralized exchanges (e.g., NYSE, AMEX) must be registered with the SEC by rule of the Securities Act of 1933. This requirement for SEC registration is commonly called "Registration Under the Securities Act of 1933."
Third: The purpose of requiring registration of stocks offered for public sale, even proprietary stocks (which will no longer be proprietary once they go public), is to provide full disclosure of financial conditions to investors.
Remember that to go public, private companies must meet stringent conditions before an initial public offering, one of which is that they become public companies, thus they cease to be proprietary companies.