Why does my nose run on one side when I eat?

This is a very interesting question, and I am afraid that it illustrates some of the pitfalls of answering questions over the internet instead of in person: there is nothing that is ever as good as a physical exam and the ability to ask further questions. Briefly, I would be interested in knowing if you have any allergies that you are aware of, and if you have any other symptoms (such as anosmia, or the inability to smell), and if this is getting better/worse. If the answer is that it is always just the one side, you have no allergies and no other symptoms, than it is likely just an anatomic variation or some odd quirk that happens. As your nose and sinuses are so intimately connected to the rest of your upper aerodigestive tract, it is possible that some of the movements that you make while chewing improve the drainage of your sinuses on that one side. So when you chew, all of the mucus that has been waiting all day to drain finally has a chance and makes a break for it. Unfortunately, it is impossible to say without a physical exam, and a quick endoscopy of your sinus (quite quick and painless, performed in the office of any otolaryngology-head and neck surgeon) would likely be the best --and possibly, only-- way to completely explain your symptoms. Please see an ENT for a complete answer, if there is one.