Thank you for the feedback, we'll take it into consideration.
With a premium subscription on Windows, you do get access to your detection history; not exactly what you asked for, but maybe you'll find it useful.

"Access your data for all websites" is indeed a required permission, and I do understand your concern. On Firefox, we don't collect any data. Full privacy policy is here: https://www.malwarebytes.com/privacy/policy-for-browser-guard.

If you are technically inclined, you can see all the requests Browser Guard makes by doing the following:

1. Navigate to about:debugging#/runtime/this-firefox
2. Next to Malwarebytes Browser Guard, click "Inspect"
3. Click "Network" in the new console that opens, from there you can see all network traffic Browser Guard is making.