A topic archive is the list of topics available through the web interface.
A group does not have to have a visible archive: it can be made visible only to group administrators and moderators. The archive could be visible only to the group members. Or it can be made visible to anyone who can see the group exists (which could be visible to the world, or only to group members).
A group owner controls the archive visibility in the group settings.
The archive can be searched.
Having found a message on the web, you can get a link to that topic to share with other users. This doesn't let you break the privacy controls: the link only shares the topic with other Topicbox users who are authorized to see it.
You can reply to a topic from the web.