Initial reports from Cuba said a key communications tower had been brought down on the south coast, interrupting electricity and phone services.
The BBC's Michael Voss in Havana says there has been torrential rain and uprooted trees but the country has a well-drilled civil defence programme and there are so far no reports of fatalities.
In the Cayman Islands, schools, offices and businesses were shut and some people were moved into storm shelters.
Property was damaged but there were no reports of any injuries.
"Our indications are that there has been minimal if any damage on Grand Cayman," said Donovan Ebanks, chairman of the Hazard Management Committee.
"We really got very luck on this one," said one resident.
But damage was more severe on the small island of Cayman Brac, to the east of Grand Cayman.
District commissioner Ernie Scott told AFP they had suffered "island-wide damage".
"Probably 90 to 95% of homes and buildings have been damaged. Some have been totally devastated," he said.
Paloma is the 16th storm of the Atlantic storm season and the eighth hurricane. It earlier brought heavy rain to parts of Honduras and Nicaragua.
Cuba has yet to recover from Hurricanes Gustav and Ike, which struck on 30 August and 9 September respectively, causing billions of dollars worth of damage and destroying hundreds of thousands of homes.
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