I can speak from experience on this. I have a relatively mild peanut* allergy - by relatively mild I mean I don't have to carry an epi-pen and I've never had any life-threatening reactions though they can be very unpleasant.
The first/most common reaction I get is a scratchy throat which results in coughing. I don't find it hard to breathe beyond the coughing but my throat feels very irritated and slightly constricted. The best way to describe it might be like feeling incredibly dry. This reaction is usually pretty quick and if I feel it coming on I know I'd better stop eating whatever is causing it. It's not fun but if the coughing isn't bad I don't always have to excuse myself from social situations and it gradually fades over the course of an hour or so. But sometimes the onset is more gradual, perhaps due to lower concentrations. In this case if I ingest enough I can end up curled up in a pretty miserable ball of the worst stomach pain I've ever felt.
Bendadryl and time help. When I was a child in school I usually carried a benadryl capsule with me, but now I'm just cautious. I'm grateful that my reactions aren't life-threatening and also that I have no problem being around other people eating peanuts, but they can certainly be unpleasant enough for me to read ingredient labels closely and ask when offered things like cookies/brownies/etc.
*known allergens - peanuts, walnuts, almonds, hazelnuts; known non-allergens - cashews, pistachios (both of which I love)