I'd go with Michael here. For all practical purposes you do not digest the gum at all. It is any of a number of mixed of usually rubbery polymers, and it passes ot through the usual channels, along with other adventitious indigestibles, such as cellulose, chitin, collagen, and lignin. Did you think that everything that you fail to digest stays inside you forever? If such things interest you, you might like to know that the content of ancient privies is of great archaeological importance. They tell us not only what people had been eating and failing to digest (hulls, fibres, and husks) but also what kinds of parasite eggs had been passing out, such as hookworms, flukes and so on. What future generations will make of lumps of chewing gum in our own sacred relics, I prefer not to speculate.
It generally is safe to swallow your gum as long as you do not swallow pieces in such rapid succession that they can meet and accrete, forming an occlusion. If in doubt, dispose of it genteelly and considerately. Say on your bedpost overnight, as discussed by Lonny Donegan.
Fascinating stuff to brood on of course, say while munching on a blackberry or occupying the throne...
