I'm not too sure about iMessage (I was a kid when it first debuted in UAE).

But I do remember them having an outright ban on BBM (for about 3 days) till Blackberry had a local datacenter (and I believe provided the govt access to the messages - I am not sure about the details in parenthesis).

Or for that matter, HTTPS?

iOS devices sold in UAE don't have iMessage and FaceTime support. I can't remember whether I was able to use iMessage the last time I was there with an iPhone I purchased elsewhere.