Click under Watchers and choose the users to be added as watchers to the task. By default, the creator of the task is the only watcher. To remove a watcher, hover over the user's avatar until an x appears and click it.
Users will need view and edit permissions in tasks to perform this action. Depending on the user's project permission settings, they may be able to view all project tasks and all project members, or just a subset of both. Watchers specifically added to the task will be able to view it.
If the user you would like to add does not appear in the list, please make sure the user and their organization has been added to the project.