From the Chat Button edit screen (Setup->Customize->Live Agent->Chat Buttons & Invitations):
Note that "Custom Chat Page" does not have an input box. If I set up a Force.com Site, I get the input box and can assign a VF page from the site. I am specifically trying to avoid the need to set up a Force.com site. Now, note the end of the introductory text:
If you do not have a Force.com site to use for these customizations, you can manually modify the chat button code on the next page.
While that is true for the images, it doesn't seem to be the case for the custom pages; in my case the Custom Chat Page. When I do configure a Force.com Site and select a Custom Chat Page, the generated HTML/JS on the "next page" (view page) include no reference to the custom chat page, by name, url, or SF ID. Is this just a mistake in the UI message, left over from when only the images could be overridden?
I understand that a custom chat page would normally be hosted via a publicly accessible Force.com site page, but in my case, I'm implementing Live Agent chat for users of the org to chat with other users of the org - no non-authenticated user will ever use chat. The OOB chat window works, but I need to customize the chat window. Am I stuck creating a site?