E.D.
I have a 2 1/2-year-old with peanut allergy, and I've known for about 2 years. Her father has the same allergy and it's quite severe in his case.
This allergy is known to get progressively worse with each exposure, so not having a severe reaction doesn't mean it won't happen later. About 80% of children won't grow out of a peanut allergy. Your doctor may tell you to invest in an Epipen to be on the safe side.
I would say avoid peanuts, peanut flour, peanut oil and things processed with peanuts. Peanut particles are easily airborne and can contaminate things in proximity easily.
Personally, I have noticed a reaction from foods processed on the same EQUIPMENT, but haven't noticed a reaction from foods processed in the same FACTORY. Most advice says to avoid both though.
You really can't be TOO careful. Other people (in social situations) won't get it right away, since this isn't very common.
Best of luck! Feel free to send a message if you have any particular question.