Do private sector organisations that carry out public functions have to publish equality information about their staff?

No. Private sector organisations that carry out public functions must comply with the general equality duty under s.149 of the Equality Act 2010, but they do not have to comply with the specific public sector equality duties. Therefore, there is no specific duty on them to publish equality information about their staff. Private sector organisations should still gather such information in relation to their public functions, to enable compliance with the general equality duty, and to demonstrate such compliance.