The prospect of future competition is not an issue. Afer all, the proprieter already faces that prospect with any of his current employees, some of whom may have worked there for years and presumably know a lot more about the business than you're liable to learn in a week or two. Also, at least in New York, if it's the corner pizzeria and not some high-falutin gourmet place, his super-secret-if-I-tell-you-I'll-have-to-kill-you dough recipe is probably the generic NY pizza dough, the same as the guy's on the next block. And the next... and the next...
The real problem is, presumably, the proprietor, presumably, has a sufficient number of experienced employees. Why spend the time and energy to train a new guy who, in a couple of weeks, just when he has learned the ropes, is going to leave?
Which is not to say that it cannot be done. My suggestions are:
1. Patronize the place you have in mind a lot. Buy a lot of pizza. Get to know -- and become friends with -- the proprietor and, preferably, the employees, too. In other words, become what we New Yorkers call a "regular." Then make your proposal.
2. Offer (preferably after completing suggestion #1) to pay the proprietor to teach you how to make pizza, perhaps before or after hours. Perhap offer to help around the place, too. ("You want to pay me to work here?" "Yeah!")
3. Even better: Do you know any other pizza enthusiasts in the area? Perhaps, two or more of you could convince the propietor, for a price, to teach a pizza-making/business class before or after hours. If the proprietor is up to it, and you get enough people to participate, at a sufficient price, maybe you could get your lesson for free.