A Watchful Eye is a Pointed Cap witch with the duty of overseeing ateliers too far away from the Great Hall or other witch societies to be monitored effectively by the Knights Moralis and other witches. They help to ensure that forbidden magic is not being used, as well as ensuring the safety and well-being of the apprentice witches. If anything wrong or illegal goes on, it is their duty to report it to the Knights Moralis.
They are not responsible for teaching the apprentices within the atelier they monitor, and they may reside in a separate building from the one in which the master and apprentices reside in.
They seem to have some form of jurisdiction within the atelier they monitor, though this is speculative.