Xylitol might give you a problem if you're sensitive to FODMAPS, I think. Anyway, I'm not sure how it's significantly different than any other sugar-free gum, plus the whole sweet tastes giving you cravings for more sweets issue.
Then again, I won't pretend that I'm always 100% paleo, so whatever floats your boat!
I know I am dragging up an old post, but just bought a box of 12 boxes of Peppersmith off Amazon. I work in I.T. and go through a pack of Extra a day and thought that I am possibly ingesting a lot of nasties. This is my 'best of both worlds' approach. Natural gum is quite different to chew and breaks down a lot quicker, but this is of course because you are not chomping on a piece of plastic.