American-educated and trained surgeons in Dubai - job offers, salary

I think you need to look at this in the proper context.

1) Yes, the salaries are attractive but they have to be considering the cost of living.  Vs. the US, you will pay enormous amounts for rentals, the schooling is what you would pay for private schools in the US etc.  Also the salaries are not as attractive as they used to be, say, in 2007/8 before the first crash.  They trended up in 2012-2013 and then the economy tanked again - see comments below 
2) As US nationals, you don't benefit from tax-free salaries as expats from other nationalities do.  So in addition to dealing with higher cost of living, you will also be paying taxes on your income in the UAE which may make the burden very high
3) As a lawyer with that particular expertise, your only options will be in multinational firms which provide services to companies across the region until you get some years under your belt.  You won't be able to represent in courts due to lack of local licensing as well as the fact that all proceedings / court documentation is in Arabic

Lastly, with the impact of falling oil prices and the resulting economical impacts; be aware that employers are tightening their belts.  The competition is now versus talent from wider middle east / sub-continent; also very well educated, close to the local culture and much cheaper than a westerner.

I am trying to give you the proper context i.e. how things have changed over the last 4-5 years in this part of the world so that you manage your expectations and do the search accordingly.

I will leave the recruitment agencies and medical field questions to people who have personal experiences in that area.