Thank you for your comments. Here's the matter in a different way. I am primarily interested in the way the question is posed. When I wrote that private messages can be read only by those who have the key to decrypt the database (i.e. a rather limited number of people) and quoted the words of theymos, the esteemed Lucius responded by saying that I cannot read the forum and quoted the words of the esteemed suchmoon as a refutation of my words.
That is, it turns out that the esteemed Lucius is 100% sure that the BPIP extension can read private messages. That's what I propose to prove in practice the words of respected suchmoon, which are so zealously referred to by respected Lucius. Because a man should be responsible for his words. I think so.
As others already pointed out, the extension has the permission to do it, which you grant to the extension when you install it. Something along the lines of "access any data on bitcointalk.org", meaning as you browse bitcointalk.org with your browser (logged in or not) the extension would have access to anything your browser downloads from bitcointalk.org. And the extension makes use of this permission to inject snippets of HTML - such as links to BPIP/loyce.club/ninjastic.space, merit counts, etc - into bitcointalk.org pages as you browse them. It does not collect your PMs though. You can verify that by looking at its source code, or you would have to trust the developers, which is what I was attempting to say in that post that Lucius quoted.
I don't know the full context that made you create this thread so let me point one other thing that is blatantly obvious but doesn't appear to be stated in that quote: the extension works only in the browser instance where it is installed and enabled. If you don't have it installed (as is the case for 99.9% users) then you don't have the above theoretical risk.
Thanks for the comment. I have your extension running and it has been granted all the permissions it requested.
Can you answer me specifically, you can use the BPIP extension to read my private message. What permissions do I have to grant the extension to do this? I want to make sure that theymos words are not outdated and still valid. And to convince the esteemed Lucius of this.