I can only speak for western SD, particularly the Rapid City area. OC is not illegal in SD; there are no state statutes addressing OC, just CC and specific places that weapons in general are not allowed.
SD is very gun friendly in the sense that are very few restrictions aside from the federal background check. Many people own guns, they are easy to find and buy, and several well-known ammo manufacturers are based here.
However, CC is much more common than OC. Aside from myself, I have seen only three others in the last few years OCing around Rapid. Many locals I've talked to know that OC is legal, but just don't care to do so. I don't really know why; South Dakotans are some the the friendliest and most polite people I've ever met, they may just be adverse to drawing attention to themselves or appearing to be confrontational. Also, many people who hold CCP's chose not to carry, OC or CC, because they perceive there to be very little violent crime in this area.
I have not had any encounters with the police while OCing. I have had a few traffic stops while CCing. The cops I've dealt with have been exceedingly polite and professional. I know from personal communication that the chief of the RCPD is aware of the Open Carry advocacy movement nationwide, and that police are taught that OC is not illegal. YMMV
Rapid City is actually fairly diverse, and many student and professionals have moved here from Denver and the coasts. If you get some bug-eyes or MWAGs, it will likely come these folks, or from the LEGIONS of tourists that we get in the warmer months.
"West River" refers to the state west of the Missouri River, "East River" to places east. There are differences in politics, perception, industry, etc. between the two, with West River being more conservative/libertarian and East River being more democratic/liberal. Some say "West River is cowboy boots and wranglers and East River is overalls and straw hats."