A quick update:
I installed ejabberd on another machine and everything (including group chat) seems to work now. I'm not sure what's changed, but I don't have time to investigate this further.
I'm still having problems with group chat using Openfire (version 3.8.1 for Windows) on this new machine.
For reference, my successful setup is running ejabberd 2.1.11 on Windows 7 (64-bit), downloaded from the link in the previous post. The firewall is disabled on the server.
In the absence of other information, it seems ejabberd is a viable choice for a chat server with Dashboard. YMMV, of course.