Android: Plenty of services claim to be able to make your messages disappear or that your conversations are private, but Surespot actually delivers on those promises. All of your messages are encrypted, and you control whether images are sharable or not. If you delete a message, it's deleted from the recipient's phone too.
Like most alternative texting apps, Surespot only useful if your friends are using it too. You can encourage your friends to use it by pulling up a QR code on your phone that they can scan to download it and instantly add you to their friends list. All of your messages through Surespot are sent with symmetric-key encryption (256-bit AES), and the keys are exchanged when you first add a user to your friends list.
You can read more about the technology behind the app here. Surespot is open source, and its code is available to view (or fork) over at GitHub.
Of course, there are ways around the protection any app offers, but Surespot has gone the extra mile to give you and friends who use the app a secure way to chat privately. When you revoke images and messages, the recipient can't see them anymore.
You can "lock" images so they can't be saved outside of the app too. Sure, you can always take a screenshot or turn off data before a message is revoked. And all bets are off if someone has physical access to your phone. Even so, if you're looking for a privacy-focused texting replacement, Surespot is worth a try. It's free, ad-free and light on required permissions.
Surespot (free) [Google Play]