More data: The prompt for username and pw occurs when I try to open the computer icon for the networked computer, so if I go to network places and double click on the "computer B" from computer A, it prompts me from username and password to connect.
However, if I type the IP address like \\192.168.1.43 it shows the contents of the networked computer no problem, and I'm able to map a network drive to a shared resource.