The Custom Contact module creates a custom user contact page using Views. The Flag module is used to mark which users to include on the page so the views can include any arbitrary collection of users. The Draggable Views module is used to allow you to drag 'n drop the entries to arrange the list in any order.
Drupal does not ordinarily allow anonymous users to access personal contact pages, so the Contact Forms module is used to create contact pages that can be accessed by anyone, and we use the Flag module to toggle on and off the option to make the contact form available to anonymous users.
The User Management page is available as a tab on the User list, at Admin >> User Management >> Users >> Custom Contact Page. On that page users with permission to administer custom contact page can toggle options for each user to add them to the Custom Contact page.
Users with permission to administer the custom contact page can also toggle those options on the user edit forms.
You can edit the Custom Contact page and the management view at Admin >> Build >> Views.
You can edit the Anonymous Contact and Custom Contact Page flags and flag actions at Admin >> Build >> Flags.
Note that links to the contact page will only be created for users who check the box to allow access to their contact page on their user edit form. The user decision to hide their contact form takes precedence over everything else.
If you want anonymous users to be able to see user profile pages, be sure to give anonymous users permission to access user profiles at Admin >> User >> Permissions.
If you are allowing anonymous users to access contact forms, be sure to use spam protection on the form! Modules like Captcha and Mollum can be used to protect the contact form from spam.