At the risk of repeating everything that's in the above links, here's the general story.
For the most part, the state owns the land underneath navigable waterways, from shore to shore between the mean high water marks.
Practically speaking, that means that it's OK to drop anchor in a river, and it's OK to wade down the river within those bounds. Many landowners do not understand that, and problems do arise. It's easy to see their point, however, since there are just enough litterbugs out there to make a mess of almost everywhere.
That said, this is only a general rule. In some instances the state has indeed sold the ground under the river, too. An example that comes to mind is the Icicle River near Leavenworth. Where the river borders on private property, the owners own the ground to the middle of the river. Wading or dropping anchor would be trespassing. However, they don't own the water, so drifting or rowing is OK.
Technically speaking, you'd even be trespassing if your bait or weight hit the bottom, but I've sure not heard of anyone complaining about that one.
The technicalities and the specific laws and legal principles are pretty well discussed in the links above.
Fish on...
Todd.
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Team Flying Super Ditch Pickle