Is the Everything2 chatterbox a private chat room?
Literally anybody can register at Everything2 and thereby gain instant access to the chatterbox. It is accessible to all, therefore it is public.
Everything2 has an XML ticker which continuously outputs the content of the chatterbox. The ticker is publically accessible, even to unregistered guest users. Though it is not possible to speak in the chatterbox through this medium, literally anybody can read the chatterbox - it is a "goldfish bowl", not a closed room.
Since the XML ticker is public, it is possible for anybody to use the ticker to log everything that is said in the chatterbox, and publicise this log in turn. Ascorbic's Chatterbox Archive does exactly this, though ascorbic himself is a registered user who is cooperative with Everything2's administration. There may also be other, silent chatterbox archives.
New users of Everything2 see a link in the Chatterbox which leads to the Chatterbox help document, which explains all of the above. This document is also publically available to guest users. The existence of the Archive is public knowledge.
Assuming that all new registered users read the help document, it is probable that most registered users are aware that everything they say is being logged publically.
Finally, this is the internet. It is reasonable to act under the assumption that everything you say on the internet will ultimately be seen by every human being in the world, including your S.O., family, employers, pupils etc. To act otherwise is irresponsible and naive. There are no private chat rooms.
Does the Everything2 chatterbox give an adequate impression of being public?
I don't think so.
Guest users visiting Everything2 are not able to see the chatterbox nodelet, nor are they informed about the existence of the ticker.
It is true that the Chatterbox help document is public, but it is unlikely that a guest user would have had occasion to read a help document on a subject s/he is not even aware exists. So, I think it is probable that most guest users are unaware that the chatterbox exists, let alone that they can read it freely.
In order to access the chatterbox, a "gate" - user registration - must be passed. In addition, since it is possible for the administration to exclude registered users from the chatterbox, the user must also, to some extent, secure the approval of the administration in order to be allowed to
Excluded users are not permitted to read or speak in the chatterbox.
Unregistered guest users are not permitted to read or speak in the chatterbox.
The Chatterbox help document is linked from the chatterbox itself, but it is reasonable to assume that most people do not read help documents.
While it is sensible to be paranoid on the internet, there are circumstances which can fairly be considered safe. For example, when using a webmail service, it is reasonable to assume that the contents of your emails will not be read by the webmail providers. If a website has a private internal chat room which appears to be only visible and accessible to registered, approved users of that website, it would be reasonable to assume that only other registered, approved users of the website are reading and logging what is said.
Is this a problem?
I think that the current situation, in which the general public has free access to things said in what superficially appears to be a private chat room, is a problem.
If it is agreed that this is a problem, there are some possible actions.
Make it more obvious that the chatterbox is a "goldfish bowl"
Warnings not to share private information and/or a visible permanent link to ascorbic's Archive would help. Also, making the chatterbox nodelet visible (though not usable) for guest users would make this more "transparent". It could also potentially encourage more users to register.
Disallow public access to the XML ticker and ascorbic's Archive
I am told that the XML ticker was originally private, but public access was opened up in 2002, to allow ascorbic's Archive to work properly. Closing it again would break the Archive (though it could be repaired) as well as almost all third-party chatterbox clients (some of which would not be repaired). It would require a lot of work. Allowing Everything2 users to use the same credentials to log into ascorbic.net would also require a lot of work, all of it by ascorbic. This is not a step which could be taken lightly.