Confide is an iOS private messenger that's like Snapchat in a suit
Confide users connect to each other via their email addresses. This is to ensure that users who don't have the app installed will be notified about a message they've received through it. Confide's creators also thought of a way to circumvent preserving self-erasing messages by screenshot-taking. Users must pass their fingers over the text to reveal parts of it, which means that screenshots won't include the whole picture. The app also notifies the sender if the recipient took a screencap.
Of course, a service which encrypts and erases sensitive information can easily turn from a solution to a problem itself. As Confide never stores a copy of the information passed through it, denying that a particular message ever existed is basically a no-brainer without the possibility to prove otherwise. Unfortunately, Confide's originators can only go so far as to insist that its up to the users to abide by respective corporate rules regarding internal communications.
If you think Confide could be of use to you, the app is available for free from the App Store.