Under Wyoming law the islands, the banks and even the bottom of the river are private property. Unless you are in a section of the river that is clearly public you are not allowed to touch bottom. This includes dropping an anchor or standing up in a float tube to pee. You are not even allowed to retrieve an object dropped overboard if it is touching the bottom.
The island at the confluence of the Encampment and the Platte is private property, owned by the private ranch across the river, it is patrolled and they do not take kindly to trespassers. Found out the hard way when I stopped for a pee break a few years back, guns and a sheriff were involved, made for a stressful day.