Usually, a director of a company is not required to hold any shares of a company. As an incentive to become a director, that person might be allocated shares in the company. Your sentence should not be influenced by whether the company in question is a listed company or a private company. A director can be given shares in the company regardless of whether that company is listed or not. As a director, that person should not be expected to assist the CEO, so your assumption that the person is expected to aid the CEO is incorrect. Hope this helps.