There is a popular belief among people concerning chewing gums or what some people call bubble gums whereby they believe that our body can't digest chewing gums and they stay in our stomach for years.
If you always swallow the chewing gum in your mouth after chewing it, here is a good news. Don't worry because it will digest normally along with your other food.
So this belief is totally wrong but in all cases, what's the purpose of swallowing a chewing gum? It has no benefits and no nutrients to supply your body with so it's better if you throw it when done chewing.