Do both Kamui Sharingans share one dimension? Since Kakashi has the left eye of Obito, and Obito has his right eye intact, after awakening the Mangekyō Sharingan, they both gain the ability of Kamui in their eyes.

Using Kamui, both are able to teleport things to the other dimension, though Obito does it far more easily than Kakashi, initially. My question is, since they are the same pair of eyes, which once belonged to a single person, do they share the same dimension? 

Say, Obito captures someone and keeps them in the other dimension, can Kakashi bring them back, as they may share the same other dimension?
The Kamui's alternate dimension is one and the same for both Kakashi and Obito. This has been seen a number of times during their fight in the 4th Ninja War.


Kakashi teleported Gedo Mazo's neck using Kamui, only for Obito to promptly bring it back using his Kamui
Kakashi got a clue to Tobi's identity because a kunai he teleported left a scratch on Tobi's mask who had teleported himself at the same time to avoid Naruto's Rasengan.
Kakashi then teleports Naruto's Rasengan so that it can hit Tobi in the alternate dimension where he just teleported his arm.


However, the first clue to this fact was given much earlier when Tobi came to talk to Naruto. Kakashi tried to use Kamui on him, only for Tobi to remark, "Don't bother with that Kakashi. It won't work on me."