If the Oort cloud extends half way to Alpha Centauri, wouldn't this mean that the Centauri system's cloud would intersect ours?
It might, if the Oort Cloud extends that far out. Current theory states that it extends out to nearly one fourth the distance to the nearest star.
The Oort Cloud is an extended shell of icy objects that exist in the outermost reaches of the solar system. It is named after astronomer Jan Oort, who first theorised its existence. The Oort Cloud is roughly spherical, and is thought to be the origin of most of the long-period comets that have been observed.
The existence of the Oort Cloud is theoretical, so what is actually happening in regards to the structure of the Sun's Oort Cloud and Alpha Centauri's Cloud is unknown. If it extends to half the distance between the Sun and the closest star, then indeed there might be some interaction and intersection.
However, current theory regarding the Sun's Oort Cloud is that it may extend up to 1/4 of the way to the closest star http://space-facts.com/oort-cloud/