Water use, like everything else in the US, is a patchwork of Federal, State, and Local regulations plus quasi-legal muscle. Multiple overlapping laws and agencies can apply at each level. Here's some examples in my home state of Oregon.
In Oregon ORS 537.110 flatly states "All water within the state from all sources of water supply belongs to the public." and Weise v. Smith (1869) upheld that "Every person has an undoubted right to use a public highway, whether upon land or water, for all legitimate purposes of trade and transportation." Case closed, anyone can use waterways, right?
The Lake Oswego Corporation likes to claim it owns Oswego Lake and the Lake Oswego City Council claims it can bar anyone entry. While the corporation does own the lakebed, the Oregon Attorney General says they don't own the water, it's owned by the public and the public has a right to access. Lake Oswego Corporation says that only applies to navigable waterways and according to the Federal Water Resources Development Act the lake is not navigable. Police have made it clear they consider the lake public and are not going to pursue charges against anyone using the lake.
This whole mess is currently being adjudicated by the Oregon Supreme Court. If you're interested in the legal issues surrounding public waterway use in the US, that case will provide ample reading.
While the Rogue River lies entirely inside Oregon, portions are managed by the US Forest Service, US Bureau of Reclamation, and the US Bureau of Land Management (BLM). Each have their own rules. For example, the Wild section of the Rogue River is managed by BLM and requires a permit during certain parts of the year.