HIBP enables you to discover if your account was exposed in most of the data breaches by directly searching the system. However, certain breaches are particularly sensitive in that someone's presence in the breach may adversely impact them if others are able to find that they were a member of the site.
A sensitive data breach can only be searched by the verified owner of the email address being searched for. This is done via the notification system which involves sending a verification email to the address with a unique link.
There are presently 74 sensitive breaches in the system including Adult FriendFinder, Ashley Madison, and others.