I found Boiling River by Googling for "swimming in Yellowstone." This is not a natural solitude type experience. You won't be the only ones there. While it does not show up on the official Yellowstone map, lots of people seem to know about it.
Other reviews will tell you how to find it. Read them. We missed it at first. Once we found it, we parked and walked the short trail (in bikinis and swim trunks.) Change before you commence your walk. When I first spied the swimming hole, however, I was immediately disappointed. It looked like 200 people were sharing the same secret feature that I thought I had discovered.
Oh well, onward.....
As others have said, Keens or sneakers are a great idea: you will be wading on grapefruit-sized stones. You will want swimming suits for sure.
The Madison is ice cold; Boiling River is boiling hot. Finding the perfect mixing spot is half the fun. I will say that the farther downriver you float, the more the two waters mix. If you lounge against the rock dam at the very end, the mix is ..... perfect! Oh, my gosh, I wish I was there right now! You relax in that perfect-temperature water and look over at the Madison, flowing rapidly by. What a crazy spot!
So what about those other 200 bathers? Well .... I gained many new friends! We arrived at 6:00pm and left @ 8:00pm. At least half of the bathers were foreign. There was a strong smattering of Yellowstone employees. We floated and chit-chatted with fellow tourists, seasoned employees, and people who spoke different languages -- and we decided that it really was like a giant hot tub. The social part added to the total experience! We loved it! We did not want to get out.
Depending on what you're looking for, this could be a real pleasant surprise for you.
By the way: the other two places we swam at Yellowstone this summer: (1.) The swimming hole @ Firehole Canyon drive ..... This is a huge crowd scene, but it is fun enough. Jump into the river above the little canyon and float down through. (2.) At the end of Fountain Flats Drive, there is a parking area for El Caliente Spring and the Fairy Falls Trail. a few hundred yards up the trail (which is really a road), at El Caliente Springs, a bridge crosses the Firehole River. On the left side of the bridge -- the side opposite the Springs -- we found a really cool rock basin in the river: hot tub size. We relaxed in that basin for a hour, as the water of the Firehole streamed in and through. (Maybe the Firehole was just WARMER this summer! It was lovely.)
You can get wet in Yellowstone ..... You just have to look for the opportunities.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.