Trouble using connection string to authenticate on SQL Server

The following connection string works fine when I'm already logged in to a computer on our intranet:

"Server=sname;User ID=my.name;Database=DatabaseName;Trusted_Connection=true"

But from home, using a VPN connection into the intranet, the connection string fails because I'm obviously not "logged in".

I tried the obvious approach of adding my password:

"Server=sname;User ID=my.name;Database=DatabaseName;Password=secret;Trusted_Connection=true"

But this doesn't work. For future reference, I normally log on to a computer named
"Corp123" with a domain of "corpdomain".

I'm using SQL Server 2008, and Visual Studio 2010 to perform the connection using C#.

If source is my connection string, then the following is the code I use to log in from C#:
               SqlConnection conn = new SqlConnection(source);
                conn.Open();
                SqlCommand cmd = new SqlCommand(@"StoredProcedureName", conn);
              ....


I appreciate any help anyone can give. I believe, most likely, that I have the permission to access the database as long as my connection string contains authentication information. But if it's likely I need special rights, please tell me what specific rights I need to ask for from my database adminsitrator. (My DBA's head pops off in response to vague security requests).