My objection has always been to the word "private", the use of which creates some legal expectations and requirements, depending on your country and/or state.
If I had to come up with a term, I think I would go with "offline discussion". There is already some lexical support for the term in the phrase we often hear in meetings, "Lets take that offline", meaning lets not tie up the entire meeting with something only a few of us are concerned with.
If a thread is an "online discussion", then what we think of as a PM is certainly an offline discussion.
They key to acceptance of any new term is how well it works as both a noun and a verb. We're all familiar with "PM me" or "Friend me". Seriously, "friend" as a verb?
So how well would this work? Clearly "OD me" isn't going to cath on anytime soon, as in as long as man walks upright. But what about "Send me an offline"? Maybe, but stills sounds clumsy to the ear. "Offline me" sounds too much like a request for digital euthanasia.