Yes! I do it.
But it slows my game down noticeably. Any dealer who takes the time to learn to count can do it, but it's not commonly practiced. And I don't practice doing both at the same time, so I'm not so quick with the cards when I'm doing it. In any case, it's not hard, because the dealer can still play the entire game and keep good table security. He only needs to be able to recognize cancelling cards at a glance, just like you do, and counts cards as they go into his discard rack.
How to tell if your dealer is counting? First, his game will get a lot slower, because he'll only count if he thinks you're doing it. (In my case, I count because if I think you're counting and you're tipping, I'll move the cut card deeper for you. I know you'll notice and I could use the money. It won't be anything spectacular, because there's still bigger people watching me, but if you carefully ramp your bets instead of jumping the pit won't really know you're there anyway.) Secondly, look for a dealer who cleans up his rack after every hand. Keeping the running count in your head isn't easy, so I move around the little plastic dividers in my rack to keep track between hands. When you jump your bet, I just glance down and see if the count is high. I don't need to know exactly what it is or convert to the true count. You're already showing me what you think it is.
If your dealer is doing this... forget about counting! You don't need to, and you can play a much better game. Just watch his rack and he'll count for you.
Good luck kiddies, and remember that a counting dealer is probably a counting player too, and you're not at odds with him.