As for the work permit - are they saying if a local can do it, someone cant get a permit?
Yes, that's how I read it. Best that you check the situation out with the immigration authority.
If that is the case how do so many nannies go to Dubai, Moscow etc?
They aren't in Barbados! Different rules, different places.
If you are not on Barbados, you may find that there are different immigration rules for your island.
How did your friends get a housekeeper? If their housekeeper is from England, then maybe your friends can help with how to get a work permit. I suspect the housekeeper isn't from England though.
Some info here on visa types. Not sure it's of help... though there is an address at the bottom of the page for the Chief Immigration Office, Barbados.
I don't live in the Caribbean and don't know anything about living or working there. All info I get is via Google Search. So given that you do live there, please do contact the relevant authorities in your country to establish what rules and regulations there are with regard to employing someone, in particular to employing someone from another country.