But both of those prices are always shown and you don't usually have to be a member of anything to get them as the clerk will just use the generic code. At most self checkouts there is an option for forgotten card which lets you bypass the need for an account.
There already are private prices -- the result of targeted coupons that certain customers get. Those coupons are tied to the SuperSaver(tm) account and aren't transferable.
Right, those won't go away. These offers will likely be in addition to those and because they're more targeted they will be higher value. CVS already has specific offers that you can only get in their app with your specific membership, Target uses their Circle membership to accomplish it.
The medium to deliver this customer experience are out there right now, but they're all working on adding signals to inform which offers you get.