In the case of my youngest daughter, our family doctor thought he spotted the sensitivity problem, based on her breathing difficulties (she was two years old when the problem emerged.) He referred us to an allergist, who administered what amounts to a patch test, where diluted amounts of a wide variety of substances are "pricked" into the skin.
The reaction is visible for those substances that are the most dangerous. A serum is then formulated and regular injections are scheduled to try and reduce the sensitivity to whatever substances have been identified.
The sting allergy often is the result of a single sting, where the person does suffer breathing problems. Stings are somewhat different in that they develop high-intensity reactions quickly. An allergist can spot this sensitivity through the same type of testing.
My allergist gave me one of those tests and said I was allergic to peanuts - and yet I'd eaten them for 40 years and enjoy them. I've never had a bad reaction to them. But now that I know they'll make kids walk around with little allergy kits --- hmmm maybe this is all just a potential revenue stream for them.
Maybe somebody is selling short in "Big Peanut".