People are freaked out about swallowing gum, and it's time to reassure panicked gum swallowers. I can see why someone might worry, especially a parent of a young child who has just swallowed their candy cud, or why they would tell them harmless lies regarding the chewy stuff, such as, "Don't swallow your gum or it will clog up your stomach."
But, as these things tend to go, harmless lies have evolved into pseudo-knowledge about the dangers of swallowing gum. So yes, gum stays in your system, but it also exits, too.
The Bottom Line
Swallowing gum, unless in extreme amounts which are nearly impossible to achieve, is no big deal.
Yes, gum is unable to be dissolved in our stomach acid in the digestive tract, and its strong molecular bonds keep it intact as it passes through us. The components of gum do not stick to our intestines or anything silly.
But that's all it's doing--just passing through, according to this article by ABC:
"So, if you ignore your mum and swallow your gum, peristalsis takes over. It carries the undigested mass of gum down through the oesophagus, into the stomach where it churns for a few hours, then along the intestines, into the rectum and out into the toilet."
If you're more of a visual learner, this video from Buzzfeed should clear things up for you.
As far as tales of large wads of gum being surgically removed from patient's stomachs, there is a wealth of direct information out there, but all indirect references to such rare medical cases affirm they are extreme, marginal instances.
One good bowel movement and your gum will be out of your system, even if you swallow large amounts, although why would you swallow large amounts of gum, we do not know.
In other words, go ahead and swallow your chewing gum. Unless you're worried about choking, as we all should be, in which case your gum-swallowing should be handled with caution.