The river is public in the middle to lower regions. The upper regions it is little more than a stream with barb wire across it.
You may not shoot out of the river because that would be trespass by shot. You would also need to make sure you are outside of any city limits and are launching from a ramp that allows firearms.
I would call the game warden for the section of river you plan to hunt and get their perspective since that’s who you’ll be talking to if a landowner makes a call.