The main obstacle with, "and it be as if I was on the one with HDSDR running" is, for that to happen you would have to be streaming I/Q to the remote PC and decoding it directly with HDSDR, and be able to send HDSDR control data back to the host PC.
Otherwise, you could make do with a combination of Windows Remote Desktop and if you are good with hardware and audio settings, make it play audio on the remote PC, or, you could use Unreal Media Server on the host and Unreal Media Player on your remote PC. It's almost realtime audio. There are others. Teamspeak, etc. type apps, or even Skype. I believe Logmein also does remote audio.
The true remote SDR experience will require you to use something like SDR-Radio or the WebSdr software if you can get it. http://www.websdr.org/background.html
If you are using the Softrock at 48kHz, and you have a good home network, you MIGHT just be able to get away with using Windows Media Encoder, VLC, or similar to stream uncompressed PCM audio from the host to the remote PC for direct decoding with HDSDR. However, you would still be back to the need for away to send the control data back to the host PC, thus, Windows Remote Desktop.