For some reason I read gum as gun. That would have been an entirely different thread and answer.
So what I was going to say was: Doing donuts while dumping into the air and then dispose of the gun in a fiery pit of awesome.
But that doesn't really make sense in this case... you should probably just spit it in the trash can.
1) Pretend to wipe you mouth with a napkin/handkerchief and deposit said gum into it.
2) Nonchalantly, put the gum between your teeth wit you mouth closed, then delicately pluck the gum between thumb and forefinger out of your mouth in one motion. (Flick the gum at a passing child or dog.)
Did anyone who wasn't a kid believe that thing about gum sticking in your stomach for seven years? If you need to get rid of it there's little harm in swallowing it, and if you don't want to do that just put it in a wrapper or tissue and throw it away.
I don't know if it's the "polite" way to do it, but I almost always put it in the original wrapper/tissue/paper towel and throw it in the trash. If absolutely none of those options are available, I'll step away from people and find a trash can and spit it out.
I spit it on the street once I'm out of the building. I don't chew the gum while talking to people. If it happens that I forgot to spit the gum and I really have to get rid of it, I will be a diplomat and take distance from the trash and spit it then I applaud myself and go back to my seat.
Who carries a paper napkin around? I certainly won't ruin my handkerchiefs with it!
Who carries handkerchiefs around? Are you 80 year old?
You wouldn't believe how useful they are.
Wiping prints from a weapon, cleaning bloody hands...
I'm going to just take a stand and say that there is no harm in making it into a ball, taking it from your lips with your index finger and thumb and just dropping it into the nearest trash can. 95% of the things you touch every day are more dirty than that gum ball, and if someone looks at you funny, tell them the keyboard they are using most likely has fecal matter on it.
People, please...stick it under a desk like it says in the Constitution. Or the Bible. It's in there!
Dropping it on to a sidewalk, preferably just outside the entrance of a building on a hot day is also acceptable protocol.
Placing it on a bus seat is also an alright option.
Ever heard of bringing along disposable gloves on your criminal endeavors? Get with the times grandpa, jeez!
I usually don't chew gum at work and if I do, I spit it out when no one is around or when I use the restroom. It's not that big of a deal to me because most people chew gum anyway and seem to understand. I use those strips that dissolve on my tongue if I'm worried about bad breath after lunch.
I don't swallow gum, either. It's not even the "7 years" thing; it just doesn't feel right going down my throat. I never imagined people did on purpose.