The first step to answering this question requires defining the term "private corporation." Many times, the term "private corporation" refers to a privately-held company that is either a sole proprietorship (one owner) or a partnership (multiple owners). Other times, it refers to a business that's actually incorporated under state laws, but not traded on any exchange or by over-the-counter market makers.
Why Private Companies Cannot Issue Convertible Bonds
In the instance of a truly private company that is owned by one or multiple people, convertible bonds cannot be issued. The reason has less to do with any laws against privately-held companies issuing bonds and more to do with the fact that no shares of stock exist into which to convert the bonds.
On the other hand, a closely-held subchapter S or C corporation, which does not trade on any exchange, theoretically may issue convertible bonds if allowed by its corporate charter and state laws. The feasibility of executing a bonds issue of this kind is another matter, however, because many closely-held corporations might have only 100 shares of stock outstanding, if not less.
It is not unheard of for an owner or local investor to lend smaller corporations money in the form of bonds that come with a convertible feature. However, this usually is carried out as a means of protecting the lender by permitting ownership in the company if it fails to repay the loan.