Greta. Google tells me that an esthetician is a beautician; is that right? Being a British colony, Cayman doesn't license beauticians. It does give them Work Permits to work for companies, but there is nothing automatic about the issuance of a Work Permit. The best thing you could do is to visit Cayman as a tourist and look around to see what's on offer. And, check out the online website of "Cayman Resident". Are you specifically interested in working in Cayman, or just some Caribbean island?