Summary: "private pages" are a very popular request, and we plan to implement them. However, even current "public" pages remain private until you share a link with somebody and it appears on the public internet.
There isn't a catalog of Brick pages. The only way somebody can find a page is if you share its public link with somebody.
When you share a page with somebody, they will see its children and its parents until the top-level page. They will not see other top-level pages. From this point of view, every top-level page is like a small anonymous website, independent from all the others.
One of the founding ideas of Brick is that pages are anonymous by default and can not be traced back to their owner. It is also impossible to figure out if two different pages belong to the same account or not.
If you don't give somebody a link to a page ("Copy public link"), the only way they can find it is by guessing the 12-character page ID.
There are 4812 ≈ 1.5×1020 different page IDs. If Brick users publish 10 million pages in total, and an attacker can bruteforce a billion different IDs, the probability they find even a single page is ≈0.000067.
Google only indexes pages that were explicitly submitted to the index, or pages that are linked to from other public pages.
Google does not index pages merely because they were opened in Google Chrome (or sent via Gmail, or shared in a messenger app).