Thank you for your help so far -- caliber-server now works with authentication on both my local machine, from remote machines, and from my Android's browser.
It does not work from either WordPlay or from Aldiko. Both refuse to authenticate.
I'm using a URL of the form
http://example.com:1234 - I have tried it with and without the /stanza portion of the URL, but of course my issue is authentication.
I'll try their forums next, but I was wondering if I'm overlooking something obvious.