There is always someone out there looking for a way to squeeze a few extra dollars out of gullible consumers. Tire shops offering nitrogen filled tires are no exception.
Some racing cars use nitrogen in their tires, as do some aircraft and freight trucks. If the big boys are doing it, maybe it's a good idea, right? Well, it may be a good idea for them, but that doesn't mean that these benefits will transfer to your street car.
Compressed nitrogen doesn't expand and contract as much as compressed air due to temperature fluctuations, which makes it useful for racing with its extreme tire temperatures, but not nearly so useful for road cars. Some vendors claim better mileage with nitrogen in your tires. Increased tire pressure does indeed create less rolling resistance and thus higher efficiency, but since nitrogen expands less than air when your tires get hot on a long trip, you'll see less improvement with nitrogen. Only if your hot tire pressure is at the absolute max will you see mileage improvement using nitrogen, and then only when the tires are cold, and it's doubtful that this improvement would be enough to be measurable. Certainly any fuel savings would not come close to the extra expense of using nitrogen.
Aircraft and long-haul freight trucks fill with nitrogen because they keep their tires for several generations, retreading them whenever they get bald. Nitrogen oxidizes and deteriorates the inside of the tire less than air, useful when you keep large, expensive tire carcasses for a long time. Since you probably don't retread your street tires and keep them for multiple generations, this is another benefit that is lost on road cars. In any case, the atmospheric oxygen and ozone outside your tires is chewing away at the rubber, so it would be like wearing battle armor on half your body.
During a fuel crisis, vultures are always out there trying to appeal to your fears and offering miracle cures. While the benefits of nitrogen are real for specialized applications, there's usually no plausible justification of the extra expense with your average road car. Be skeptical.