Well, a green screen would do the job, but it isn't really necessary.
Like what Endeaver said, it would be easier, cheaper, and faster to just stick your camera on a tripod, lock it down, and then tape your talent on either side of the room.
If you do need to have the clones touch, the easiest way to do that is to follow what Endeavor said as well. Also, if you decide to have the clones touch, you don't need to have someone who looks just like your actor. You can just use a CU (close up) of someone with hands similiar to your "cloned" actor touching the "cloned" actor.
There is a lot of "movie magic" out there, and many great concepts to use.
Try practicing the scene or just the concept of "split screens" or "cloning", to try and understand it better. Be creative and open minded.
We can't really describe it that much further, it will probably help you the most by simply trying it out!