I suspect the law depends both on the river and the state.  I have heard stories about my great-grandfather whose property was bordered by a stream.  He and his neighbor on the other side of the stream had a running battle.  They would put debris in the river in an attempt to change its course.
The property line was the stream and if the stream changed, they ended up with more/less property dependng on how it changed.

I think you need to err on the side of respecting private property rights, especially of the owners have marked the property as such.  If unsure, assume it is private and move on.