Nassau, yes, Coco Cay not really. In Nassau they will take US currency and credit cards. In Coco Cay, you'll be using your shipboard card. If there are any locals outside the fenced in area, they will take US currency.
Coco Cay has a small straw market and hair braiding area staffed by locals and they take dollars. As Penny said, you use your cruise card for all other purchases since it is Royal Caribbean's private island and really an extension of the ship.
When I was in Nassau back in 2007, a tour guide said don't bother exchanging your money. When I was there, $1 USD was equal to $1 Bahama Dollar. Both are widely accepted.
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2007 - Majesty of the Seas
2009 - Enchantment of the Seas
2010 - Freedom of the Seas