I need clarification: Network vs. Homegroup

I have a desktop and a laptop both on the same "Homegroup". I can access a document that is stored on desktop from the laptop, work on it, but it won't allow me to save the changes back to the original document. I am running Windows 7 Home Premium on both computers.

What am I missing?

Tech Support Guy System Info Utility version 1.0.0.2
OS Version: Microsoft Windows 7 Home Premium, Service Pack 1, 64 bit
Processor: Intel(R) Core(TM) i3-2125 CPU @ 3.30GHz, Intel64 Family 6 Model 42 Stepping 7
Processor Count: 4
RAM: 6030 Mb
Graphics Card: Intel(R) HD Graphics 3000, -1988 Mb
Hard Drives: C: Total - 936741 MB, Free - 879151 MB; D: Total - 16897 MB, Free - 2104 MB;
Motherboard: Foxconn, 2ADA
Antivirus: Microsoft Security Essentials, Updated and Enabled