The organization I work for supports projects with several local NGOs. The ones listed below are ones I have found to be great in terms of administration and finances, as well as attending to their beneficiaries' needs.
-Caritas (although depends greatly on the diocesan branch)
-CENSEL (Centro de Servicio Legales para la Mujer, based in Santo Domingo, working with survivors of dmoestic violence and all gender-based violence)
-Accion Callejera (Santiago-based, works with street children and out-of-school children to get them back in school)
-Servicio Jesuita a Refugiados y Migrantes (technically international NGO; based in SDQ; working with all migrants and refugees, mostly- but not limited to- Haitian)
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Have not worked with but have heard good things about:
-DREAM
-Mustard Seeds Communities (I have worked with them in Jamaica, so I can't comment on their operation here. However, I know them to be extremely dedicated and take in the kids that NO ONE wants, especially handicapped who have been abandoned or families who can't take care of them.)
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As far as helping, it varies from organization to organization. These days, most organizations need cold hard cash more than anything- donating money, in my opinion, is the best way for people to help these local organizations. Help organize a fundraiser among your expat community.
However, I realize that is not always satisfying to the person donating, so I would suggest that ex-pats living here- not tourists- can best contribute to strengthening the sustainability of these organizations, especially if they have a professional background that might be of use to the organization. For example, if you are an accountant by trade, help the organization set up an effective accounting system. If you have computer or marketing experience, help design a web page.