I peel the vine off with an axe, or needle nose plyers if it's small. When I split the round, I split off the outer bark where the "hairs" remain. Then lay the thin splits side by side way back in the woods for a couple of years so the rain hits it. When I am sure there are no oils remaining, I use gloves and feed it to the stove. I take no chances with the stuff. It hits be badly. And as someone mentioned, you don't want to inhale the oils if they are burned.
From my experience, the larger vines seldom give offshoots on the ground after cutting it on the tree. I was always worried about spraying the cut area, but I no longer worry about it. The smaller vines, that's not the case. It will continue to send shoots similar to a locust tree.
I have read from several medical sites that exposure to the oils over a period of years will make you more prone to breakouts. Not sure if that's true, but if it's true, that would suck.
By the way, there has been a wives tale going around for years that the ooze coming from the sores on your skin will spread the outbreak to other areas of your body, or other people. That is not the case. The cause of spreading comes from not addressing your clothing right away. My wife has gotten it several times by shoving my jeans in the wash. The oils are wicked and stay around for a long time.