This is generally the term in the States, and this "private law" area governs non-profit organizations, not "non-state" organizations.

You get lots of hits under this for Colombia, Chile, etc, such as:

www.angelfire.com/id/pccm2/ - 12k


"La Corporación de Desarrollo Económico Parque de Ciencias Cajón del Maipo, o Corporación PCCM, es una corporación de derecho privado, sin fines de lucro, cuyo objetivo es:"

or this one:


www.imagine.com.co/aida/informacion.htm



NATURALEZA
ACOLDESE es una Corporacion de Derecho Privado, sin ánimo de lucro, de carácter nacional y autonómo, de duración indefinida, sujeta como tal, a las normas consignadas en el título XXXVI del libro Primero del Códugo Civil y demás disposiciones vigentes.

XXX is a firm specialising in private law.

This removes the ambiguity. I would use firm rather than corporation. One normally talks about law firms rather than law corporations!

If you do a search on this, you will find several usages similar to the following:

"Free Europe, Inc., was established in 1949 as non-profit, private corporation to broadcast news and current affairs programs to Eastern European countries behind the Iron Curtain. The Radio Liberty Committee, Inc., was created two years later along the same lines to broadcast to the nations inside the Soviet Union."

Definitely it is a non-profit corporation, and non attached to government institutions.