In the mobile settings, I'm able to leave the 'desktop url' entry empty and I can surf the site without redirection on my localhost. However, when I put it on the live server, it picks up that _MJ is not defined and doesn't allow me to surf the site with a desktop browser.

Why the difference? I need to access some pages of the site from a desktop browser. I've looked and tinkered with the template menu settings and it doesn't seem to help.

Empty Desktop URL leads to incorrect redirect url like "http : /// index.php" and maybe to some other issues.

sorry, i don't really understand what this means.
is this a bug which needs to be fixed?
is there a hack I can apply in the meantime?
thanks

It means that Desktop URL should always correspond to actual site URL. MJ needs it do distinguish mobile (sub)domains and main domain.

If you automatically synchronize local and live websites, just exclude administrator/components/com_mobilejoomla/config.php from the list of synchronized files.