The same way coupons work in drug stores already; there's a stated price and then there's the price with SuperSaver™ membership, which you have to pull out of their app and could be subject to any number of personalizations, real-time or not.
Every time I check out at a Kroger store I tell the cashier that I would like to get a membership card. They don't require you to give a phone number until you register the card after the checkout - which I don't do because I already got the benefit of the card at checkout. This won't work at stores that require phone number first, and still leaves open credit card-based tracking - for that you actually lose the few % cash back if you want to use cash instead.
I've been using the same un-registered Smith's (Kroger by another name) card for 5 years now. I get gas points and everything. Now, do they know a lot about me based on my credit card? Yes, but it's a start...
I have one of those too.. but only used cash with it. I have heard that if you use that and a CC or debit card they link the data - even if just once - and can link the past data as well.
I also have my wifi turned on my phone so they can't correlate the 'burner card' with that.. if I don't have cash I use a different discount card.
Of course if they scan the face and correlate that way - meh. For that reason I only use cash at Target - but not sure how long that will be enough to keep from data points being put together there or anywhere at this rate.
Every time I check out at Kroger (or similar membership-based store) I tell the cashier that I don't want a membership but I do want the listed sale price. They always just scan their own card.
CVS started to refuse to scan the 'house' card as of about a year ago. and they try to collect your phone number when they hand you a card for the first time.
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.
[1] https://getpocket.com/en/