I really react strongly to mosquito bites, I get big itchy welts that last for a long time; I think that it is an allergy. My husband gets bites but they itch for a short time and then fade quickly.
A lot has to do with whether there is a stiff breeze to keep the mosquitoes down IMO; but of all the Caribbean islands we've visited, I've had the least problems with them in the Caymans and we have been there 6 times. The Caymans are a wealthy country with a high standard of living and they spend money on their islands and tourist industry.
I have seen planes spraying for mosquitoes over Grand Cayman and I have never had bad problems on the sister islands either. On Cayman Brac I have gotten bitten up while visiting the caves and the lookout points in the marshland, but no real bug problems around the resort or beach or just traveling around the island and stopping at various sites.
Also, the Caymans have a low incidence of mosquito-borne viral infections, like Dengue and Chikungunya, while nearby islands like Cuba and Jamaica have many more cases; another indication that they take mosquito-abatement seriously.
Of course, it is the tropics and there are mosquitoes and you will probably get some bites, so bug repellent is a good idea. I recently found a new product that helps a lot after you get bitten. It's Allegra Anti-Itch ointment and it works well to relieve the itch and it makes the bites fade and not come back.
Have fun on LC!