Hi @xsans
Welcome to the Power Pages forum.
1) I work with local users. I can register them, but then the login doesn't work? What should I do?
Ans: Local login is not recommended by Microsoft. Please use Azure AD B2C. You can register and log in as a user.
Official link: click here

2) I add a list of products, how can I do it so that users can EDIT the records?
Ans: For Power Pages, Contacts is the key. Assing an appropriate web role to the contact, set the table permissions for your list, and add your web role.
Official link: Click here

Hope it helps.
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