Introduction / Needs / Structure / Jobs / Money / Why?
Needs
An Island
At least of the ones I've seen, the most affordable ones are in the Philippines, but I would hope to find one offered by Tonga or other Oceania island nations, because they seem to be both stable and have a good system for making island leases. Most leases are for 99 years, and prefer some sort of positive economic impact on the nation. That is one reason we will have local employees.
A good island will be well above the sea (typhoons aren't fun to watch from underwater), ideally with some cliffs for climbing, diving, and windmills (for electricity). It should have a decently-protected moor for a boat, and a place for a small amphibious plane to be stored in case of a storm. A beach would be nice. Vegetation is also a must. Because there will be a fair amount of people living on it, it must also be at least 20 acres.
It's also important that we check the place out before buying it, of course. It might have pretty pictures, but it could be a mosquito-infested swamp rife with diseases that we thought had been eliminated by modern science. Proximity to a hospital should also be a consideration.
Buildings
With the purchase of a boat, much of the shipping of raw materials can be done by the group, and assembled on the island itself. Many home kits are available, but the exact logistics of construction depend greatly on the proximity of the island from proper civilization. This is by far the least-investigated part of the Paradise, Inc. project so far.
The buildings would need to be able to put up with very extreme weather, such as typhoons, and be very energy efficient, since we would rely on solar and wind power. Lighting would probably be with LED lights, and any other means of cutting corners would be taken.
That said, solar and wind turbine power have come a very long way, and should be able to run even high-power things like the water treatment, computers, and stereo systems without a problem.
Waste compost, water reservoirs and reclamation, ducts and piping would all have to be fairly top-notch to make it as efficient as possible.
Food
There is a goal of a self-sustainability with this project, so much of the food has to come from the island itself. This is partly practical and partly ideological. Fortunately, many foods can be grown, and we will keep a few chickens (for eggs and meat) and a goat (for milk and all its derivatives). Cows are a bit too big for a place with poor refrigeration, and require a lot of maintenance. Goats basically take care of themselves.
Some essentials can be imported, such as flour, oils, and some spices. Fortunately, these items can last a long time and aren't needed in vast quantities.
Most of the gardening and cooking responsibilities will lie with those who love gardening and cooking. It's a practicality thing, you know? We hope to find some people who want to grow stuff for the rest of their lives, and don't mind farming a little extra in exchange for living in paradise.
And yes, there would almost certainly be a distillery.
Water
Desalinization has come a long, long way. For about $10-20k you can get a salt-water purifier that's about the size of a dorm refrigerator, and it will give you 1,000 gallons of potable water per day.
We will need a clean reservoir, a means of getting the ocean to the filter, and a pump. Rainwater collecting has also worked well for humanity for... ever.
Transportation
A good-sized boat is obviously necessary, especially if we plan on shipping building materials. The boat also can move more passengers, is more reliable than a plane, and less harmful to the environment.
It wouldn't be hard to make the boat self-sufficient with today's power generation (solar, wind generators), and desalinization processes. A 60' ketch should be capable of sailing most anywhere, including crossing the Pacific if necessary, but I am waiting on an expert to be sure of that.
Making a 'green' aircraft presents a more formidable challenge: you can mix the fuel (20 bio/80 regular), and lower emissions by 50%, in order to use a typical amphibious like a Cessna 208, or hope that the DA42 Twin Star gets a water-landing version with more cargo room. You'd have to get the plane from Japan or the Philippines if you wanted to have an island in the Pacific, though, because crossing the Pacific from the United States requires a larger craft than the company really needs.
It's a ridiculous notion, but I'd also want to turn my Baja bug into a biodiesel schwimwagen for diving trips and cruising around the island.
Naturally, while the island will be nice and all, people will also want to leave to visit relatives from time to time. These modes of transportation will make that a little easier. Hopefully, each person will eventually earn a pilot's license to make the logistics easier, as well.
People
While I wouldn't want to pigeonhole people into jobs that can't be replaced (in case they want to leave), the company would definitely give each person a job. Ideally, each person would really like their job, and since we would all essentially work for each other, there wouldn't be a need for prescribed hours, per se. We'd do what we needed to do, help others when it's needed, and go do whatever as individuals or groups.
Of course, I'd want to be sure to bring people we all want to live with. People will have their own homes and all, but community meals are going to be a big thing (since it's shared food), so picking your friends will be important. Also, divvying up expenses will be a good thing since maintaining the boat and plane won't be free.
Fun
A wide variety of activities would be available; I have a hard time picturing people getting bored. Apart from the obvious things to do on a tropical island (diving, surfing, scuba diving), many other things can be done.
If there are cliffs (and there probably would be since one requirement of the island is that it needs to be high off the water), you could go rock climbing. Indoor activities, like board games, and other tournaments could be organized, especially for the kids.