SimpleX Chat
Messaging with no user identifiers at all, not even a random number.
A private messenger keeps your conversations between you and the person you are talking to, not the company in the middle. The bar is end-to-end encryption that is on by default, not buried in a setting. The harder problem is metadata, the record of who you message and when, which is often more revealing than the words. The apps here are judged on how much of that they encrypt, collect, or throw away.
Strong encryption is close to standard now. The bodies of your messages are usually safe. What separates these apps is the metadata around them, like whether you need a phone number to sign up, whether your contacts are uploaded, and how much of who-talks-to-whom the servers can see. The best are built to hold as little of that as possible.
Here are some things to look for.
Messaging with no user identifiers at all, not even a random number.
The mainstream secure messenger, trusted and easy to use.
Onion-routed messaging with no phone number and no central servers.
Open, federated chat on the Matrix network, no phone number needed.
Paid Swiss messenger with no phone number or email required.
A hardened fork of Signal for Android, with extra at-rest protection.
Encrypted chat over ordinary email, with no new account needed.
Anonymous desktop chat over Tor with no servers or accounts.
Bluetooth mesh messaging that works with no internet at all.
Serverless peer-to-peer messaging with a cryptographic ID.
Distributed peer-to-peer calls and chat, a GNU project.
Peer-to-peer messaging over Tor that even works with no internet.