Stores is built to know as little about you as possible. You sign in with Apple, so there's no password and no email list. Stores never asks for your contacts, there are no ads or trackers, and your data is never sold.
Some things never leave your phone. The camera reads barcodes and photos on your device, and only a barcode's number is ever looked up. The home-screen widget reads a local copy of your list and makes no network calls. Location reminders, when they arrive, will run on your phone too.
Want to start over? You can delete your account and everything in it from Settings.