Signal works with a contact list, for private 1 to 1 and private groups. You need a phone number to use it, and they claim to encrypt almost all metadata, such as message senders. Signal claims they cannot read the content nor the history of users actions (but you have to trust them on that, they claim to use SGX enclave stuff, but can anyone technically verify that they do what they say they are doing?)
Element works with rooms, there is no contact list. A 1 to 1 conversation is a room with 2 people. The homeserver Matrix.org stores all metadata and they are readable (metadata are not private/not encrypted) by Matrix for some features to work. You only need a nickname to use it (at least for now). The content of conversations are e2e encrypted. In theory, i understand it would be possible for a matrix server to delete any metadata/messages once messages are delivered, but some features would not work, and you would also have to trust the server to actually delete the metadata.
Would be happy to read anyone who could correct or complete me.