Are there any side-effects of whitening your teeth with baking powder?

Doesn't really . Baking powder by itself doesn't really whiten teeth. It is a mild abrasive which sort of scrubs off stains or "sands" them off. This only works on the surface if teeth but doesn't get in any cracks or imperfections if your brush can't get in them. Best to just use a bleaching product that you get from your dentist or from a drug store if you can't afford the dentist's.
Virtually none. Baking is found in many natural toothpastes. It can be used as an effective teeth cleanser, particularly when used with dilute peroxide. However you must wash your mouth with water afterwards to avoid getting a good dose of sodium!