Can't say for sure about SD, but I did have a conversation with a CO in ND about it a couple of years ago. Long and short of it was that once another person leaves and his birds are still with you, they are now part of your possession limit. That was in ND, but I'm guessing that's about how it would be in SD too.
The reason I was discussing it with the CO was that for the past few years, my wife and I would drive up to ND in Sept for sharpies/huns. She would hunt a week and then fly back home to TX. I would carry on hunting another week or so.
I have a 12v fridge/freezer with me on all hunting trips, and my vacuum sealer. Other than the birds that we cook and eat while on the trip, I clean them and vacuum seal them (leg or wing attached) and then put them in the freezer. Last year we ordered some insulated freezer bags from amazon. They look like a padded envelope, silver/foil on the outside. When she flew home from ND, we put 7 or 8 frozen/sealed sharptails in her suitcase amongst her clothes. When she got home it had been around 8 hours since we packed the suitcase in Bismarck, and the birds were still frozen hard as a rock. They hadn't started thawing even a little bit. So that worked like a charm.