Like what Mr. Thing says.
I've been here less than a year, so still recovering from the start up costs, but €1500 per month is about what I have budgeted for savings. That's based on the wife not working, no kids and a pretty decent 2 bedroom apartment in the Marina costing around AED 140k per year. We eat out once or twice per week and don't buy into the 5-star Dubai lifestyle.
As others have said, it all depends on the sort of lifestyle you want here. A single person, living frugally, could probably save €2-3000 per month. If you get blingy with your accommodation, car, shopping, brunches etc, and especially if you have a wife and kids to support, you could quite easily blow the lot.