I just spent a week in Montana fishing some of the known waters. Never got close enough to another human to speak. Did see one or two people within several hundred yards.

My answer to the original question is, "it depends". On a popular water like the San Juan, your personal space is about thirty feet around you. On the lower Madison, if someone is within a couple hundred yards on my side of it, I think they might be too close, especially if they can clearly see that there is nobody else for miles. My local water, used to be empty most weekdays, now is much more crowded. I have a cars-in-the-lot limit I turn around and go home when I reach.