Local and/or real name of this tree.

That looks like Tan Tan to me.

Leucaena leucocephala is known in the Virgin Islands as tan-tan or guinea tama- rind. Native to Central America, it is an erect woody plant that grows as a shrub or tree. It tends to have a single central trunk, without major side branching. In the Virgin Islands it usually reaches be- tween 15-20 feet tall.

It's not tan-tan. Locally called woman's tongue or similar due to the rattling the seed pods make in the wind. 

Here you go: 

Albizia lebbeck, known as Tibet, Mother in Law's Tongue, or Woman's Tongue Tree is naturalized in the Virgin Islands. It is originally native to southeast Asia, New Guinea, and northern Australia. It is very noticeable during this time of year, the drier season, as it sheds its leaves and the dried seed pods are prominent. They are very noisy trees when the wind blows and the seeds in the dry seed pods rattle, hence the name Mother in Law's Tongue or Woman's Tongue! The wood from these trees is valued by wood-workers for the contrast of the light grain and darker streaks.

yep CruzanIron is right. I couldn't tell without any greenery on the tree or much to show the scale of the tree, and I forgot about the Mother in Law's tongue!

+ 1 for @cruzaniron and the Tibet tree

I've also heard it being called the rain tree because it sounds like it's raining when the wind blows and turns green again quickly when the rains come.