N.B. This answers the question of how one can prevent two miktex installs from occupying space with duplicate files. For a Windows/Linux dual boot environment, one should look at using texlive as per the instructions linked to by Gonzalo Medina in the comments to the original question.
I guess it should be possible to achieve what you want by creating a symbolic link to the path where you installed the repositories.
I haven't tried this, but I think the following should work:
- Install miktex on both operating systems. I assume you have installed under c:\program files\miktex in both cases, and that each OS's C drive is mounted as D:\ on the other OS.
- Delete the c:\program files\miktex folder in operating system number 2.
Open a command prompt in OS 2 and type
mklink /D linkName c:\program files\miktex d:\program files\miktex
This should cause OS 2 to behave as if OS 1's install of MikTex were its own.