Ia Orana,
I am an American who retired in Papeari, Tahiti. As an EU citizen, you can come here, but if you want to live here, you will need to apply for resident status. This also includes how you intend to support yourself - in other words, you must have sufficient funds to live here and support yourself. You will most likely not be allowed to work (if you can even find a job) as jobs are first given to locals due to the very large unemployment. It would be allowed for you to start your own business. There are definitely different rules for DOM/TOMs, French overseas territories. For example, no Euros are used here - just Pacific French Francs. I hope this helps.
Nana,
Manarii Tane (Steve)