Please consider adding "advance search" to access permissions. We are setting up permissions for "visitor card data entry" and it just seems like a bad idea to allow "advance search" at this level...you really can get quite clever w/ this search function once you start poking around. :)
Thanks for the email.
I can definitely add in a feature request for this. For now though you can always Restrict what fields people can view in Advanced Search results using this article. It's also worth noting that people won't be able to bring up profiles in the results of a search that they shouldn't be able to access otherwise via their people access restrictions.
Hope this helps,