For Oregon, I know that in some spots it is legal to camp on the beach. But there's a lot of parameters involved. Basically, any spot that WILL be legal to camp on the beach is going to not be easily accesible by road, and probably would mean shlepping a loaded bike along a beach for aways, or through the woods. I think the main idea of legal beach camping in Oregon in the modern era is to give folks who hike the Oregon Coastal Trail https://en.wikipedia.org/wiki/Oregon_Coast_Trail
a place to camp every night, as there won't always be a state park campground around, esp. in the southern portion of the state. As such, the idea of the free beach camping is designed around through hikers, NOT cyclists.
Here's a good discussion about the legality of camping on beaches in Oregon:http://www.oregonhikers.org/forum/viewtopic.php?t=4900
To the OP, if you are willing and skilled in the stealth camping arts, there is nothing stopping you from doing that as you travel the coast. But most likely, you'll be camping illegally, even on the Oregon Coast.
And as others have said before, the frequency and cheapness of hiker-biker campsites along the coast means that you'll really want to be in the idea of stealth camping instead of using the designated spots. For example, I was out at my favorite hiker/biker site at Cape Lookout State Park last week. It's $6 a night (per person) and the campsite is literally a hundred yards from the actual ocean.