There are a few privileges that I'd prefer to carry over network-wide (eg vote counts). This isn't one of them.
Each site has different post quality rules, and edits must adhere to those as well. This requires a bit more participation.
Also, as a newbie to the site, you may not know of some edits that ought to be made. For example, on Mathematics.SE, Physics.SE, Chemistry.SE, etc a seasoned editor will know that posts with plaintext math (usually) are to be LaTeX-ified . There probably are other such guidelines elsewhere. So, if a new user edits/comments in the review queue, it shouldn't disappear from the queue--there may be other things to look for.
Another concern is how easy it is to get 2k and the badges on some sites. Especially SO (where nobody will notice your edits as the front page is anyway flooded). Such users may not really know what edits are for, and we don't want them to remove questions from scrutiny on other sites via the review section.
So basically, in my opinion, it is preferable to keep such privileges separate for each site.
Edit: OK, now I see what you mean--you mainly want it removed from your own queue.
As per the above half of this post, I do not advocate giving the "remove from the
/review queue for everyone". I don't support any non-read-only privileges copying over sites (except the ones that already do like commenting). Even if the privilege is rather minor like this one (my kook opinion, I guess :P)
But, I think that this would be nice--a checkbox in
/review marked "hide posts which I have already reviewed", and that hides such posts regardless of your badges. This may be a pain to implement, though.