I don't know the technical details of that for sure, but I think the answer is that keys and chat history are stored on-device only; for example you lose your WhatsApp history if you don't restore a backup when moving to a new phone.

If a messaging app is showing you message history in a private browsing window then perhaps the encryption key for that history is derived from your password or something like that; that can be done locally so that all the server ever sees is encrypted data.