You may not want to swallow your chewing gum, but if you did, it wouldn't kill you. It wouldn't stay in your stomach for seven years, either. The dangers of swallowing your gum have been greatly exaggerated.
Photo by torbakhopper.
Your body does the same thing with gum as with anything else it can't digest, like all that good-for-you fibre. You just poo it out.
So where does the myth come from? Possibly from a real but rare phenomenon called a bezoar, a mass of undigested matter stuck in your stomach or intestines. If your cat hacks up hairballs, you know what I'm talking about.
There is such a thing as a "chewing gum bezoar," but it's so rare that there have only been three cases reported in the medical literature -- ever. We'd still like to see you properly dispose of your gum, but that's because swallowing it is gross, not because it's unsafe.
What Really Happens When You Swallow Gum? [Greatist]