Here are some examples from my own experience, where I have had to use RDP:

1) Get a call from the users that the application is very slow, and I can't connect from SSMS either. It could be that SQL Server is maxed out, or the box is hung. Hopefully, the former will be limited by the resource governor or WSRM, and then I can log on to the server to investigate further.

2) Good luck loading large event or SQL Server logs over a network.

3) Let's say I have to upload MPS reports to Microsoft. Not really practical to copy huge files to my workstation.

4) If a flat file has to copied from server A to server B to be loaded into SQL Server, then I am going to have to log on to one of them directly, or everything has to go thru my workstation.

As mentioned above, these problems are most likely to occur off hours. Adding a VPN to the equation slows things down even more. And then there is the communication issue. I am already on the phone with someone for the initial problem. Now I have to call my server admin. Maybe I can use Skype for one of the calls, except that it may cause something else to time out. And will your third shift be adequately trained to support DBA's, or will they eventually have to call you as well?

So bottom line, does your management understand and accept the additional downtime and lack of productivity that will result from the DBA's not being fully able to do their job?