It's not a binary offline / online business. Retailers are a mix of both, with some, like Walmart, quickly expanding their online presence as well.
How do we make companies follow it is different issue altogether . First step is the make liability exposure high for violating (like COPA ?) and provide consumes right to access what they have on you.
There is no technical way privacy of this or any other kind can be guaranteed, companies internal controls are always going to be opaque and data collection looking from the interfaces we can audit may not visible .
We can only make it higher risk legally and give consumers frameworks to make it easier to expose violations
Aggregate data on what consumers bought from the retailer. This is the retailer's data.
Individual sales data from sellers of the retailer's platform. This should be PII and treated like any other sensitive data.
Aggregate data on what consumers bought from all sellers on the retailer's platform. This is the retailer's data.
Individual data on consumer actions on the retailer's platform prior to purchasing an item. This can / should be PII. This includes search history, shopping cart, listings visited, etc.
Aggregate data on consumer actions on the retailer's platform prior to purchasing an item. This is the retailer's data.
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My position is it's unreasonable to ask retailers to limit their use of aggregated, anonymized data. And unless you limit that use, it doesn't materially change the current situation. The major difference between Walmart, Costco and Amazon is Amazon has a lot more aggregate data.
Costco at best knows what I bought at their store, Amazon knows a lot more than that. Just from the data they collect on the store they will know how many people compared what products with what else, what are they searching for, what is in who's shopping list, they will likely know demographics and lot more information about the buyers as well.
It is not remotely the same problem between online and offline.