Almost two years ago, Twitter launched encrypted direct messages. I wrote about their technical implementation at the time, and to the best of my knowledge nothing has changed. The short story is that the actual encryption primitives used are entirely normal and fine - messages are encrypted using AES, and the AES keys are exchanged via NIST P-256 elliptic curve asymmetric keys. The asymmetric keys are each associated with a specific device or browser owned by a user, so when you send a message to someone you encrypt the AES key with all of their asymmetric keys and then each device or browser can decrypt the message again. As long as the keys are managed appropriately, this is infeasible to break.

But how do you know what a user's keys are? I also wrote about this last year - key distribution is a hard problem. In the Twitter DM case, you ask Twitter's server