Howto use Windows Authentication with SQL Server 2008 Express on a workgroup network?
Whatever username you are using on the "remote" machine must be set up on the "server" machine.

So, if you are logged-in to "remote" as "John," there must be a "John" user on the machine that is hosting SQL Server 2008 Express, and unless you are going to provide different credentials, it should have the same password, too.