The user will need the space admin permission to manage watchers on pages in his space. If you are the Confluence site-admin, you can go to the space, then Space Settings. Click on the Permissions tab and the Edit Permissions button (there are two, one to add groups, one to add individuals.)
Once a user has the space admin permission for a given space, they can add watchers to individual pages. It's not a separate permission; you must be a space admin to be able to add watchers.
From the individual page watchers management, you can also add or remove watchers on the entire space the page lives in.