I think you are experiencing the "double hop" issue with Windows Authentication (you can Google that for more details). Basically, Unless you have Kerberos enabled for these servers, you cannot pass the Windows credentials for User1 from one SQL Server over to your remote linked server.

I do not know much about Kerberos, other than it is required if you want to pass Windows Authenticated user credentials between servers.

Generally I will define a standard SQL Login on the remote server and use that login to set up my Linked Server. In other words I will be logged into the primary SQL server with my Windows login, but will log into the remote server with a static SQL login. Not the greatest solution if you need to audit the actual user accessing the remote server - but the only way to do this that I know about.