Although you can't see an actual chat window in any of Adriacraft's screenshots, you can see that there are the options to edit "toast" settings, including messages, which implies that chat, or at least notifications of messages received, could be coming soon to the desktop client.
There is also the option to remember status, but before we get excited about offline mode finally arriving, remember that there are already statuses in battle.net -- away, busy, etc. There is, however, an image of the user being offline, but this could just be because he wasn't logged in when it was taken.
If you are concerned by any of this, or feel that you would not want the friends of friends option, setting up parental controls on your account can be an excellent way to tailor battle.net communications to your personal preference. You can disable it, or even selectively disable various elements, including parental controls.