We recently upgraded one computer to Windows 8 with Office 2013 and now then that user opens up an Excel file on the network, other people no longer see the "document in use - read only" notification.  It does say that it's read-only in the title, but other users frequently overlook it.  Is there a setting that needs to be turned on so that the notification shows up?

No. That  is not work properly. In normal situations domain or Windows login name is not showing. You should to have Sharepoint to have information like that.