Cameron,
I talked with our developer who is using our Print Server Object Extension from a WPF application.   Within the WPF application, a web reference is created to our WSDL for the SOE.  They are calling it with the following code:
            string xml = Serializer.SerializeMapInfo(MyMap, txtTitle.Text.Replace("'","''"), cboPaperSize.SelectedIndex);
            using (var proxy = new LIS_Print_SOE_lis_print())
            {
                AppStatsLogger.Log("LIS", "PDF made", 1);
                proxy.LIS_Print_PDFCompleted += proxy_LIS_Print_PDFCompleted;
                proxy.UseDefaultCredentials = true;
                proxy.LIS_Print_PDFAsync(xml);
            }
I am not an expert on this method to pass the credentials, but you can see that there is a 'UseDefualtCredentials' property.     I am assuming that it is using the user's  credentials on our city domain to pass to the SOE.
Hope this points you in the right direction,
Mele