If you are writing, then you can just use the MonoGame NuGet’s after you have created your project, then they don’t need to install MG it’self.
The only other reason they would need the MG install would be for the Content Pipeline project tool, which is only in the installer at the moment (a few have asked for that to be packaged separately). So long as they don’t need to edit / add content, then the NuGet’s would be all you need.
To use the NuGet’s simply:
1: Create a project using the MG templates (which it sounds like you have already)
2: Remove the MG references (just a precaution, not absolutely necessary but recommended)
3: Right-Click references and select “Manage NuGet Packages…”
4: Search for the MonoGame packages and install the version you need for your platform
Done
Just rebuild your solution and then share the game folder. Only share the packages file in the packages folder with the solution, don’t’ need anything else in that folder. When others open your project, so long as they have NuGet installed (comes by default in 2013/2015) it will automatically download the latest packages on build.
Hope that helps.