Who would see the comments?
Their reply to your private reply should be automatically private, effectively creating an isolated thread.
People who want to disagree indefinitely on some fine point could do so without creating a ruckus.
If anything especially worthwhile came up in the discussion, either party could easily just repost it as an ordinary public reply.
To discourage people from harassing each other with private replies, you should only be allowed to downvote them.
This would also allow people an easy way to say "Thanks" or "I agree" without having those comments pollute the top level space.
News.YC does support sending arbitrary messages to other users. It's just only turned on for us and YC founders. I could turn it on for everyone. Would anyone want that?
The advantage/disadvantage of using messages is that they wouldn't be tied to a particular comment. There's also the disadvantage that everyone would invoke whatever that quote is about every system evolving till it adds messaging...