Planets sharing an atmosphere would produce a highly unstable situation. At the closest point between the two planets gravitation would be negligible and would result in huge amounts of water, foam, rock and debris being drawn up into the atmosphere.
It is likely that this region would form a rotating storm between the two planets. Given the high energy content of storm systems and the very low net gravitational forces in the area it would result in a storm of biblical proportions and would absorb any storms approaching it from either planet. No Storm would be able to cross this boundary.
Weather would be dominated by the ring storm between the two planets. The weather systems would be powered by the energy from the sun coupled with the difference in ground speed between the point of closest approach and the far side of each planet (similarly on Earth there is a difference in ground speed between the poles and the equator that drives our climate).
This would cause all of the weather systems on each planet to spiral from the far side towards the point of closest approach where they would be consumed by the world storm system circulating in that area. Similarly on Earth storm systems spiral away for the fast moving equator to the stationary poles.