It's all hit or miss... The closer you are to the "big town" hot spots of Aberdeen, Huron, Pierre, Chamberlain, Mitchell, etc. the more hunters you will see. If you can get an hour or so away from each of those spots, you're chances will increase on finding ground that hasn't been hit as hard. In general across the state, if I run into a landowner that borders a piece of public land I usually ask them how often that piece gets hunted. The answers are pretty consistent along the lines of, "it really goes in spurts. Might go days or weeks without seeing somebody here, then it might be several days in a row with non-stop hunters."
So the short answer, is it's all luck based on hunting public land that has or hasn't been hunted. You can do things to increase the luck in your favor, pieces that are down minimum maintenance roads, far from towns, a long ways off of paved roads, has good cover in the area (grass, cattails, water, trees, corn), being quiet when entering the land, not yelling at your dog, hunt a big piece of land but get to that tough to reach far corner that not everybody makes an effort to get to, hunt that small piece of cover that other hunters may overlook, hunt the piece opposite direction you think you should (birds are smart, and if hunters hunt it the same way over and over, eventually they will learn how to easily escape, so surprise them and come a different way).
I used to live near the river bluffs and would hunt them because I lived there. My experiences are they are tough walking with lots of wild flushing birds, but there are birds. Chamberlain and Pierre are your bigger towns along the river and lots of hunters also go there, so there's competition. I don't have experience once you get west of that or on Sharpies.
I do think that any one hunter can reasonably expect to get a 3 bird public land limit in most of SD if they have a little luck, decent shooting, work hard, and hunt from 10am to sunset. But all it takes is a few misses or a few hours spent in a bad area to throw those odds off pretty quick.