Ah, the ongoing dispute of riparian rights!
In general, a landowner owns the river bed (actually water course bed) to the centre of the river and, by definition therefor, owns the whole river bed if he has land both sides. You are committing trespass if you walk the river bed.
However, who has rights of navigation is a whole different matter and is in dispute. In Britain there has been practice to use rivers to transport people and goods without hindrance though, in some cases, by toll. Now, the owners of riparian rights on many rivers want to stop that and prevent the use by boats so they can make money from the fishing. Neither side wants to take it to court because neither is 100% sure of their position