How would someone go about creating something like the Pew Research Center?
If you wanted to create something like the Pew Research Center, you could just take a look at their history (that's their own, or you could check Wikipedia) and see how it was done in the first place (founded in 1990). The short version: a newspaper company wanted survey data to be able to report on, and so they created a center to do just that. It was then spun off into a charitable trust supported by various and sundry donation sources.

So if you wanted to take a lesson from their founding director, check out Andrew Kohut's history according to Pew or according to Wikipedia. Note that the Pew Research Center was a late-career project for him, and he served many smaller positions inbetween.

You could similarly check out the history of the Gallup organization (founded 1935), though a more recent example might be The Gapminder Foundation, founded by the late great Hans Rosling (he talks about his own personal life-long journey in this last book, Factfulness - a great book, IMHO).

In general, most organizations with this sort of large impact were founded by a combination of people with a business/financial interest (charitable or otherwise), as well as people with an expertise often driven by years of working directly in the field. Kohut did it serving years for survey/polling companies and had a graduate degree in sociology, Rosling did it after many years serving as a medical doctor and specialist in public health, etc. 

A research-focused graduate degree would probably be hard to do without if you wanted to be a subject-matter specialist, while a graduate degree focused on professional preparation around business/fundraising/etc. would tend  to be most helpful if you are not already independently wealthy and highly connected and wanted to be involved on that side of it. 

Another path, rather than starting from scratch, is to seek out organizations that are already operating in your country with a similar mission, or to look for organizations operating in other countries that already have or might be interested in expanding their operations to another country. Find what sort of positions they have, and then look to see what requirements they look for, so you could obtain direct experience without needing to start from scratch. If you wanted to start their own, chances are you would be competing with them directly or indirectly anyway, in terms of funding, public attention, etc. You should be able to find potential opportunities for volunteering, internships, etc., where you could be involved before you've finished your education, which would help you ensure this is what you want to do and give you a big leg up in getting started.