I just use the native Reminders app. I created a list called "Shopping", to keep things separate from my general "Stuff" list and "Bills" reminders.
As a cool bonus you can use Siri to add and view items. Tell Sir to "Add eggs to my shopping list" and they get added. Tell her to "Show me my shopping list" and she does. You can even check them off in Siri's view without opening the app.
Of course, there are shopping list apps that have more advanced features, but my needs are simple. Reminders does it 'good enough'.
As far as privacy, Apple is pretty serious about it's customer's privacy. They are not going to share your shopping list with anyone, even if you do use iCloud to sync it with your other iOS devices, Mac, or icloud.com.