Our friends over at Adriacraft have had the chance to test a newly datamined version of the battle.net desktop client. They have various in-depth screenshots on their article, but it looks like the current friends list has undergone a needed overhaul. You can now add and remove friends, report other players, accept and decline friend requests, and see recommended friends, which are friends of friends.
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.