Two options for managing watchers:
1. If you are a JIRA admin or have permission to Manage Watchers, you can click on the ellipses on the task which will open a dropdown - select Watcher menu.
2. Click the number next to Watch.
In both these cases, you will see a little box appear with the current watchers of the issue. When you have the proper permissions, this will also include a text field in which you can add users in order to make them watcher of the issue.
For availing the notification option using @ mention , the Browse Issue permission is necessary for those users.
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