It is not possible to train with a private physician. I guess it is technically possible to train, but it wouldn't count for anything. You couldn't become board eligible, accept medicare or private payers, or get insurance.
My guesses to what it means when it says 'private physician' are that it's a graduate from a while back who's working as a private physician or is someone else not currently in training- shadowing, doing an observership, or something like that. Those are just guesses. We could probably figure it out given more specifics.
In any case, to answer your question- no, you couldn't go to med school then train as an apprentice vascular surgeon with your current boss. You'd have to go to medical school, do your 5 years of surgery residency, do your 2 years of vascular fellowship, then go work with him in 12 years if you're both still up for it.