Can anyone help?

Start by making sure it isn't a scam - Way too many of them around but they only look good on the surface so are easily busted with the slightest investigation

Not a scam (100%) so please advise.

You can save on most salaries it all depends on how much you want to spend you could earn 50k aed and not save or 5k aed and still save
The good thing is cost of renting is coming down in most areas in Dubai, you can Google to give you an idea for the area you plan to live but 27k aed will more than cover the rent
Regarding your privacy no one will come knocking on your door in Dubai as long as you keep your private matters private

As far as your concerns go, read this thread:

https://www.expat.com/forum/viewtopic.php?id=818357

While the question was from the perspective of single women; my comment is on the overall aspect of doing things which are not exactly considered kosher as far as the culture goes.

On the salary, it is not bad for a single person.  You can save a lot or very little depending on your lifestyle.  Eating out a lot and partying can dent it quickly.

Thanks for the replies lads, I really appreciate it.
I'm not really someone who likes going out, I prefer sitting in and having a drink myself (If I do fancy that). I also hate eating out and cook indoors - this was also the case when I lived in China for three years.
So can I expect to save on the 27,000 per month? Is this a salary most people can or do save on?
I would like to date there - and I'm not one to cause a scene - but would I need to bring someone past my doorman?

Also, as the job is lecturing, I know how to censor myself (again, see China) but what's the general tolerance for things like LBGT among the city's young people? I have two gay friends who have had a blast out there and not experienced any problems but I'd be curious to hear.

I read somewhere the average pay in Dubai is 10K aed so yes you can live and save earning 27K aed
On LGBT again keep your private matters private
Public display of affection by LGBT will probably get you arrested

I already commented on the salary and the door man aspect; read the thread, should not be a problem.

LGBT angle is not accepted by society or tolerated by law.  So keep it under wraps and don't advertise preferences - it should be fine then.

Hi 
Regarding your salary being enough...Ill give u a breakdown of your basic expenses, if that helps.

Studio or 1 bed in the areas you have mentioned (in a good quality building) will cost you aed 6 to 10000 at best.
Add aed 1500 for the utility bills,
aed 2000 for leasing a car/ or aed 20 per trip for Uber, ( sticking to Marina and MOE areas),
Aed 50 per meal at a respectable restaurant/ or aed 2000 for groceries per month (water, fruit, breakfast and 2 meals and snacks - for a single person).
Entertainment and Furnishing - depends upon your needs
These numbers are very close to approx figures (will probably be less, particularly current rentals) and will change depending upon your lifestyle.
And you will still end up with a decent amount as savings, with the salary that you mentioned.
Cheers!