it doesn't make much sense, to keep private data, in public blockchain.

Technicly this can be done, and it was actually done: https://steemprojects.com/projects/p/steem-messenger/

but one of a problem is... that it cost money (0.001) to send a message, because in fact message was sent in transfer. Another inconvenience is if you will loose your private memo key, then you will not be able to read all messages.