I swallow chewing gum whenever I've finished it. It's the sensible thing to do. That way, it doesn't end up on the
pavement, our cities are cleaner, and you'll improve your
swallowing ability, making it easier to take pills, etc..
I'm astonished at how widely held is the misconception that it's somehow dangerous to swallow chewing gum. I've received warnings from intelligent people chewing gum "stays in the system for (insert number here) years" and that it'll "get stuck in your stomach and could kill you".
Chewing gum is indigestible, or rather the great majority of its components are. However: so is cellulose. Or lactose, for many people. Both these substances are staples of the Western diet. Indeed, the eating of indigestible materials is practically advised, as it bulks out faeces and reduces the likelihood of bowel cancer (hence the reason "high in fibre" is emblazoned on cereal packets).
But imagine chewing gum is dangerous. Can you really imagine that Wrigley's would have managed to avoid an enormous lawsuit? Picture people left in agony, unable to digest food, with swollen appendices AND CALLING THEIR LAWYER RIGHT NOW.
So, either it's perfectly safe, or there's a massive conspiracy involving the Vatican and the Freemasons (and probably the Illuminati) and most importantly Wrigley's to cover up all chewing gum related deaths. Which must be in their millions, by now.
NB - whilst I didn't actually get a consultant to verify this WU, I can claim some limited credibility insofar as The Straight Dope happens to agree with me here...
http://www.straightdope.com/classics/a2_059.html