Yes, they are all public, as far as I can tell. Their status in Toolset is “Published”, they generate working archives and everything seems fine with them. The archives are visible to anonymous visitors.
I don’t know which settings can affect it, I tried several settings in Toolset’s taxonomy screen, but the select box won’t appear.
Perhaps it is related to this?
__________________________________________________ Meta box callback function
_ Hide taxonomy meta box.
If you disable this, there will be no metabox on entry edit screen.
meta_box_cb ______________________
Provide a callback function name for the meta box display.
Default: None.
___________________________________________________
This is at the bottom of the taxonomy settings screen in Toolset.
As far as I remember, Toolset’s taxonomies are registered by the Types plugin, which is in the official WP repository, so the commercial Toolset modules may be irrelevant.
It will be awesome if you can make it work, but anyway I am very happy that it already works globally for archives.