After running in the Exchange Shell
 add-MailboxPermission "user X" -user UserY Fullaccess

to give userY the ability to access UserX's mailbox, is there anyway that User X can tell that User Y now has full access?
['https://serverfault.com/questions/244842', 'https://serverfault.com', 'https://serverfault.com/users/73606/']
This is not detectable in Outlook, OWA or any other Exchange client, which is how all standard users usually access Exchange; however, a skilled user (or an administrator) *could* detect it by looking at the actual permissions on the AD user object or on the Exchange mailbox; this requires no actual console access to the Exchange server or to DCs, it can be accomplished with administrative tools running on any client system.
The user (user X) that has had the permissions changed on their mailbox will not know unless the mailbox is opened on another users (User Y) machine, and User Y makes changes to the contents of the mailbox.

In the past I have dealt with managers who have demanded this capability, and things got really ugly when the manager was accidentally deleting/changing items in that users mailbox.