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Varies from bike to bike for me, but I'd say I'm on the drops about 1/3rd of the time - or maybe 50% of the time I'm not climbing, I like the old school racer position for anything that requires hard acceleration, braking, or technical descents because they put me in a lower position and make it easier to shift my body weight around the bike.
Lately I've been using the drops for descending less and less though though, I've been experimenting and have been surprised to find that even on what are relatively moderate descents I can go just as fast by getting into a good tuck as I can by pedaling. I have known for a long time that tucking was faster than spinning out a big 52x12 gear, but I'm finding that the break-even point for pedalling downhill for me is closer to the point where I spin out 52x14.