Martinique, Guadeloupe and half of the island of St Martin are all classified as overseas départements of France, and hence part of Europe. This is good news for holders of European passports, as these islands are simply an extension of mainland France from an immigration point of view.
Martinique has an official language of French, uses the Euro as currency, and the local population are full citizens of France with full legal and political rights including holding a French passport. This is also the same for the local populations in both Guadeloupe and St Martin.
St Maarten/St Martin is an island that is half Dutch and half French, hence the two names. The French part is a part of Europe, whereas the Dutch part, Sint Maarten, is a part of the Netherlands Kingdom, but not classified as a part of Europe; this is the same for the other parts of the Netherlands Antilles, including Curacao, Aruba, Saba and Bonaire.
EU citizens can live and work in the 3 islands belonging to the French West Indies, as if they were in any part of Europe. Taxes will need to be paid and contributions towards social care.