The short answer is no, for various reasons. The first is that atmospheres just aren't very thick, so the two bodies have to very close to each other in order for the atmospheres to interact. This will cause one or both bodies to start to disintegrate (see the Roche Limit). I don't think you realize just how thin atmospheres are. The Earth's atmosphere is 99% below 30 km, and the Earth's radius is about 6400 km.
Also, consider a satellite in NEO (Near Earth Orbit). It has about a 90 minute period. That's about 17,000 mph. If this is a near pass by an object in a larger orbit, the speed must be much greater in order to let the object get farther away from the earth. But then the orbit can't be "sharp" enough at close approach to do a (roughly) 180 degree turn - that only happens for something like LEO. Although an orbit can get closer than the 90 miles or so (early satellite orbits) the gravity pull on the orbit simply doesn't increase much.
Also, if two bodies passed close enough to mix atmospheres, the atmosphere gases would be the least of your problems. At the speed they're going, you would get really massive devastation from the shock waves - 17,000 miles plus over a HUGE area.