Security model
What Simple WhatsApp Button can read and write, what it stores, and what leaves your instance.
What it reads
res.partner: Mobile, Phone and Country. Nothing else.
What it writes
Nothing. The link is a non-stored computed field.
What leaves your instance
Nothing. The module makes no outbound connection of any kind. The link is built in Python and rendered into the page; it is followed by the browser only when a person clicks it, and at that point the browser goes to wa.me - which is a normal outbound link, the same as any other on a web page.
Access control
None of its own. Anyone who can see a contact can see the button, because anyone who can see the contact can already see the phone number the button is built from.
Personal data
No new personal data is stored, and no processing happens on it. The number is already in your database; the module reformats it for display.