as far as i know, you only share your "theory" values (which makes it eaiser to reshearch the subject in question, for example "infantry", "armour" "Light Aricraft". a high theory can cut somthing like a 1/3 off the Research time.)
however, you can ask your allies to let you build things via "production licenses" , which lets you build thier version of that thing (for example, a destroyer flotilla, or a brigade of heavy armour). you use your own IC and materails to build it, but it is built to thier tech design. this is done via the diplomacy screen (look for "buy production License"). bear in mind you can't buy every type of unit (captial ships, notabley, cannot be made this way)
so, you can't get whole techs off other nations, just quicker reshearch, but you can get the improved units via another methord.