I have never used the KME diamond stones but my guess is they are not much different than DMT stones for instance. You can use them dry I am pretty sure but I have been using my DMT stones with water and a bit dish soap. Just enough soap to mitigate the water tension on the surface. I find the feedback maybe even better this way and they stay clean. You don't end up having all that black metal dust on your fingers and everywhere else in the air. At the end I rinse them, dry them a bit and done. I do that too on the DMT dotted stones even though the dots work as a reservoir for "shavings".
I have used the DMT Diafolds too and usually dry. I have never gotten really good with them other than a applying a microbvel and I guess that is really what they are made for although then I don't understand why they come in so many different grits? You can do some serious grinding with them too and I guess that is why but since it is a handheld and difficult to lay down on a surface, my consistency lacks quite a bit, compared to the DMT credit cards for instance which are great too.. Anyway, that gets quite dusty so rinsing them under water is a good idea, unless only used for microbevel.