Twitter, by design, is a public platform for sharing up to 280 characters at a time. If you want a more private chat, then there's direct messages, but they can still be snooped on and remain stored as part of your account. However, Twitter is considering a much more private messaging mode called "Secret conversation."
As TechCrunch reports, the discovery was made by Jane Manchun Wong in the Twitter for Android APK. The feature is present, but not enabled, suggesting it is currently being tested internally and isn't ready for public consumption.
Twitter is working on End-to-End Encrypted Secret DM! pic.twitter.com/2lLr5i1p42— Jane Manchun Wong (@wongmjane) May 4, 2018