Why are mailing lists not showing in the Memberships component?

In order to see memberships the Mailchimp Memberships component, several things must be true:

• The object where the component is added to the page must have at least one email field.
• Users must have at least read access to the contact, lead, and if applicable, person account email fields.
• Users must have one of our permission sets. 
  Note: The Mailchimp Limited Access set will keep the Manage List button hidden, but users will still see the record's related memberships.
• Sharing rules must not make our custom objects, like Mailchimp Audience and Membership, private to the Security Owner.