To be honest, it's hard to define a "local". Most people you interact with, such as those in restaurants, museums etc are from all around the world - the majority from the middle east and far east, of course, but not necessarily from the UAE. As for locals wandering around in shopping centres etc, I've had no trouble.
The only trouble you do have is on the roads, where without wishing to make this sound like a slur on the whole population, they're all mental. Drivers in the UK can be pretty stupid at times, and when I went to the States the drivers on the freeways around LA were a bit crazy, but in the UAE a large proportion are either dangerous, incompetent or both.
This is perhaps understandable of a nation which has had such rapid development in the last quarter century and where such a large proportion of people have suddenly had the income to afford bigger and quicker cars, in a country where the road network isn't nearly up to coping with the quantity of traffic. Still, this doesn't excuse people darting between the cars in 60mph traffic like they're filming Fast and the Furious, changing lanes for absolutely no reason, speeding up to rows of stationary traffic and braking hard at the last minute, crashing red lights, double-parking in lanes on a roundabout and all the other stuff you see all the time out there.
I consider myself a fairly courteous driver but in a week of driving in the UAE I quickly realised that courtesy means you end up being pushed off the road. You have to adopt a very much offensive style of driving as opposed to the defensive style you get taught in driving lessons.
But yeah, driving aside, I had no trouble with any locals at all. It's so multicultural that you don't even feel like a minority if you aren't Arabic.