The company does provide accommodations in Abu Dhabi, and while you are on your two week shift you live on base (typically in the clinic). You can call it a tour, a rotation, a shift, or the 14 day countdown- it's all semantics.
While you are free to travel anywhere you'd like on your two weeks off, unrestricted, your residency visa will be for the UAE... So according to the government, the company, and your passport you live in Abu Dhabi - if found to be otherwise I'm pretty sure immigration will revoke your visa.
I'm not sure about living elsewhere on the down low. Years ago, there was a guy who used to travel to Thailand, where his family was, for everyone of his days off. I am unsure if that is allowed still, but I'm sure he still had an accommodation and address in Abu Dhabi as well.
They never made any empty promises to me.
But since this thread has been rehashed, I'll update that I resigned from the company a few months ago to pursue PA school and Flight EMS in the states. I also just finished up my associates degree.
My director, Jon, was one of the best mangers I've ever worked under. A true leader, rather than a boss. And my time there was well spent and a great experience. I made friends that I'll probably have for life, and do not regret taking the job one bit. My only regret is not being able to stay for the duration of my contract, but a shot at being a PA in a quick timeframe was too good to pass up.