I was looking through the Gramps Web features list and didn’t see a mention of (expansion on) privacy.
As a single-user application, the regular Gramps offers no options for redacting information in the Views that are marked as Private. That is only used in filtering and Reports.
Since Gramps Web is multi-user with multiple-levels of access, is there a function for masking data marked private (without the data in the served HTML) for users below an access threshold and for anonymous visitors?
You are right, this needs to be explained more explicitly (even though the word “privacy” is on the front page 
)
In fact being able to hide private data is one of the huge benefits of Gramps Web and it’s very robust and secure thanks to the way Gramps is set up: by using Gramps’ PrivateProxyDb as the only means to access the database for users without sufficient permissions, even before any genealogical or other computation is performed, hiding private data is done at the database layer, so no bugs in view functions can expose it.
Whether or not you can view private data depends on the permissions of the user, see User system. “Guests” are the ones that cannot see them.
In my own personal tree, by the way, all the relatives are “guests”, while my core family members are “members” and can view private records.
Since the removal of private records is done at the database layer, it is very thorough: the count of total number of objects on the front page, the serach function, thumbnails, also private secondary objects like names or event references - nowhere will you find a trace of a private record as guest user.