Much of the precipitation for the year for these places comes from powerful winter storms that can add inches of snow every hour to the mountains and still have enough moisture left to snow heavily in the ‘rain shadow’ areas.
Moisture is also far more likely to fall as snow in the interior, as it tends to be colder and hotter on the other side of the mountains from the coastal plains (it’s very rare for snow to fall west of the Cascades or Sierra Nevada’s)
Cold air doesn’t hold as much water. So as moist air passes over mountain ranges and is pushed upwards, it tends to eventually fall out of the air. That might not be as it’s ascending but perhaps afterwards (on the eastern side in your examples). By the time what remains spreads out in to the wider, warmer, lower-elevation land afterwards, that air is able to hold the water no problem, and it doesn’t rain.
The way warmer air holds more water and how significant it is, is why weather services give humidity as “relative humidity” (the percent of water in the air relative to how much water air of that temperature/pressure *could* hold) instead of “absolute humidity” (how much actual water a unit of air is holding right now).
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