Click on the add button under Watchers and choose the users to add. To remove a watcher, hover over their avatar until an x appears and then click it.
Users with view and edit permissions in bugs can add watchers. Depending on the settings of the user's project permissions, they may be able to view all the project members in the user dropdown or just a subset of them. The same is true for viewing the list of project bugs, but watchers specifically added to the bug will be able to view it.