Typically in a workgroup / homegroup environment network access to shared resources (files, folder, printers) are granted on the basis of usernames and passwords (not computer names).
In order for this to work as seemlessly as possible insure both computers have the user accounts and passwords configured identically on both computers then when sharing pick "share with"->"specific people" and choose that user from the drop down list.
Alternately you may also be able, when prompted for credentials, type the MachineName\Username in the username box and then the password for that user on that machine (the machine on which the resource resides) check the box that says remember credentials...