(This question was originally asked in a too specific manner, as can be seen in the revisions. I only re-asked this in a better way to answer my own in hopes it might help someone. It turned out to be a pretty simple task).
In the web.config (client and server), in the <system.serviceModel>
section add/modify a binding to look something like this:
<basicHttpBinding>
<binding name="MyBasicBinding">
<security mode="TransportCredentialOnly">
<transport clientCredentialType="Windows" />
</security>
</binding>
</basicHttpBinding>
And, add this to client side web.config <system.web>
section:
<identity impersonate="true" />
<authentication mode="Windows" />
The two changes will make the end-user the current user of the web request which will then be sent in the WCF message.
The user can then be retrieved on the server side like this:
ServiceSecurityContext.Current.WindowsIdentity