Ho do I know if I need to see a doctor after being stung by a wasp?

If you were allergic, you would have breathing difficulty immediately. Or swelling of the throat, dizziness, hives away from the sting site.
A bee or wasp or yellowjacket sting can get HUGE even on someone who isn't allergic - in a way, we are all allergic, but most of us recover with just some baking soda and water paste applied to it. A sting on an arm or leg can make the entire limb swell and not need medical attention.