One way to do this is to make a copy of the module and make the position a place on the mobile template.
For example to put the Login module on each of the mobile templates you would make 4 copies and put the first into mj_iphone_middle, the next into mj_imode_middle, the next mj_pda_middle and then mj_wap_footer. You then choose which menu items to associate with the module in order to specify which page(s) they show on.
Would this satisfy the stats program?
The reason you need 4 copies is that the way Joomla! is setup the module is positioned in the same place whichever menu (page) it is on. I couldn't find a position name that was common to all mobile templates.