This happens because the remote server is not exposed to the internet.  You will be able to access these files when you are on the local network because the requesting user has access locally.  To get around this, one option would be to create a virtual directory on your webserver an point it to the directory on the remote server.  Then you can request the files on the remote server using a normal http request.

Webserver
http://www.mywebserver.com

Remote Server
need access to remotecontent folder

Create Virtual directory on Webserver and point to Remote Server /remotecontent

You then access the files using
http://www.mywebserver.com/remotecontent/filename.ext

That depends on your authentication method.  If you're using Windows Authentication, you can do this with regular file/folder permission.  If you are using Forms Authentication, you will have more difficulty.  With Forms Auth, your users are really accessing as anonymous (or IUSR_ServerName, or however else you have it setup).  Regardless, your remote server won't know who's who.

To accomplish what you want, you have a few options.  The two simpliest (imho) would be to

1. Purchase a robust file manager like Speerio File Manager Pro and configure it do what you need at the folder level
2. Write some code that compares the folder name to the username and allow/deny as appropriate

I'm sure there are other, more elegant ways to do this, but I tend to go for what I can get working in an hour or so.  Either of these two options fit that requirement.