Where to save messages in Django app?
It depends. You could save your messages in the database if you need the classical approach to messages. This would best work for messages sent in e-mail or bulletin board "style".
If however you need real-time behavior and lots of messages, having a database inbetween might slow things down a bit. You could use node.js for a real-time chat.
If you are intending to use Django-Messages as a means to communicate between users, it wasn't meant for this. It is simply used for displaying flash notifications for users when they navigate through pages in your app. They are normally stored in the session and used only for the following page, afterwards being deleted.
I cannot give more details because your question is quite vague. Firstly you need to find which are your plans, how do you intend to use the "chatting between users".
I want to make my own on Django-app. The app must have an messages-application not only for notification but for chatting between users.
So, for example I've write a model for messages, so where I have to save the messages? In the database, or maybe it is not a good style of coding and bad for work speed?