Creating messenger for django
Or you can install xmpp server (like eJabberd) and write a server side interface over it. It will be easier, faster and optimal solution. 
Gmail and Facebook both uses xmpp protocol. People using your application will also be able to send chat request to their friends in gmail.
You wont even have to write a website interface, there are javascript library (like Converse.js) available which you can directly plug into your website and you will be good to go.
I'm a learning Python/Django programmer and want to try to create an easy web-messenger. Is it real to write web-messenger for django? And does any modules for that exist or any open-source protocols support python?