How do you whiten your child's teeth without going to the dentist?

Keep them Clean. Children need to work on keeping their teeth clean and should not use whitening products. If you are referring to a child with all perm teeth, (teenager) first again is cleaning and if you want to whiten them there are products at the local pharmacy that may help but your dentist is by far the best option.
Cause? Baby teeth should be almost paper white. If not, see dentist asap. Adult teeth are very light when they erupt. Keeping them clean is best option (brush 2 minutes 2x/day). Also need to floss daily. There's no pill, mouthwash, toothpaste, bleach, otc, food, abrasive, or homeopathic substitute for regular dental care. It's false economy.