As was already said, testing is probably merited and the risk likely depends somewhat on which allergen you're thinking of and what form it's in. But for anecdata on blowing air transferring allergens:
Many years ago a customer who made various doughs was convinced the gluten LFD strips we sold them were not working, because they got what seemed like false positives at random. It took a facility visit to figure out the issue: their QA lab had an in-wall air conditioner that blew air in from the production floor. The countertop where they would run their gluten tests was right in the path of the AC. If they ran their tests while the production floor was running a gluten product — BOOM, occasional positives. Tests while the production floor was offline or running GF — no gluten found.
If you were chilling, say, chicken and also fish filets — it would be easy enough to run some fish LFDs on the surface layer of some chicken to collect some data.