MLA’s Consumer and Patient Health Information Section (CAPHIS) outlines what it takes to be a consumer health librarian (unfortunately only in English):
The Librarian’s Role In The Provision Of Consumer Health Information And Patient Education
While not officially a set of core competencies, the document does describe the special skill set required for this type of work.
Also check out the other useful information CAPHIS provides about managing a consumer health information service (from link above see Managing a CHIS on left menu).