Do you know if we can chat with a specific user in private? (with StackExchange network of course)
No. Private messaging is not available anywhere on the network (with the exception of moderation issues).
While Private Messaging is not a feature of the Stack Exchange Network you can however make chat rooms that people can join, while they are public rooms you're find that not many people hop into these channels and you can always keep an eye on who joins should you want to discuss privately.
You can setup a Chat Room by following these easy steps:
- Click Chat
- Create a Room
However it should not noted:
All conversations here are public, and they are logged and licensed under the same cc-wiki license as The Stack Exchange Network.