It's actually the slower Pokemon that will get the weather condition. In all the games, abilities come into play AFTER both Pokemon have been released. Their abilities are also determined by speed. (For example, take Pressure, where the faster Pokemon exhibits the ability first.)
So, for example, if Kyogre were faster than Groudon, Kyogre's would use Drizzle first, after which Groudon (the slower one) would use Drought. So the prevaling weather condition would be Sunshine, not Rain.