Do shareholders need to approve a stock transfer between company directors?

This is a privately held company.

Under most state laws, shareholders have the right to approve (or disapprove) any and all ownership transfers. If the company has multiple classes of stock, the shareholders' agreement may specify that only the owners of certain classes will vote on that particular issue -- in other words, it's not an issue open to all shareholders, only some.

To get an answer to your specific situation, speak to an attorney in your state to review the different classes of stock, voting rights and corporate bylaws, all of which will have a bearing on the answer.