Permission to edit should check entity_id_str

Maybe I've misunderstood the purpose of this module, but it seems to me like a primary use of this module would be to allow group members to manage the menu items for the entities related to the group. Members should be able to add and change menu items for the nodes that those members have permission to create or edit. I don't see any permissions related to menu items, only menu entities. After turning on all permissions for group members, rebuilding caches, rebuilding permissions, I still am not able to access menu entities or items as a member (in a test environment). Is there a regular Drupal permission that I need to turn on (like accessing admin pages, or admin theme?) to allow access for members to menu entities or items?