Typically the "rivers" in Montana are navigable and that scenario does not seem to be the point of contention. A typical legal hunting scenario involves boating on the Missouri River through private property to access public BLM land for hunting the Breaks.This thread popped into my mind while having a roadside chat with a game warden here in Montana a couple weeks ago. I asked him "Would it be legal for me to hike up the side of a river going through private land, but always staying below the high-water mark, to access public land to hunt for deer or elk?" His response was that as long as I started on public or private that I had permission to be on, and stayed within the public stream access while traversing the private, I was good to go. I asked a few follow up questions, but as far as he was concerned it would be perfectly legal and he would not write a ticket for it.
The issue arises when the scenario involves Class II smaller streams or creeks someone wants to wade up to access public land beyond private property. An example of that would be wading between high water marks of Rock Creek through the Crazy Mountain Ranch to gain access to the west slope USFS public land of the Crazy Mountains. Methinks your warden acquaintance just might write you up for that one. (Especially after receiving photos and your license plate number from the CMR Marlboro wranglers.)