"Які неурядові організації ви знаєте?"

Translation: What non-governmental organizations do you know?

[PaulMetcalf]

In the UK we refer to "private sector" and "public sector" organisations - "government" and "non-governmental" sound out of date.

[Gremmie1]

This usually works in the USA as well, although private sector generally implies business for profit. One would probably specify if it is distinctly a private non-profit organization.
The phrase "non-governmental organization" sounds clunky and I don't know that I've ever actually heard it spoken outright like that. We pretty much always abbreviate it to NGO.

[Weylin366674]

I'd certainly use NGO to describe an organisation such as Médecins Sans Frontières.

[Stephen127987]

I wrote What non-government organisations do you know?