According to the episode, Will asks which one of them is real and Dr Crusher answers that they "both are". But I'm not sure if that makes the most sense.
If you take a starfish and either cut or rip it in half, the two halves will regrow a entire starfish, complete with any physical abnormalities the orginal starfish possessed. Identical in DNA too.
The Old Mixer wrote:
In "The Enemy Within", neither half of Kirk is the whole package...each possesses qualities that are integral to his personality.
An interesting detail about this episode is, according to Deanna after the transporter incident Riker changed, his romance with her became less important and his career gained importance.
While less than with the two Kirk, perhaps the Riker who beamed up wasn't quite the same Riker. There was a psychological difference.