Do both Kamui Sharingans share one 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."