Courts have determined that for this purpose the NCMEC is an agent of the government. NEMEC is 99% funded by the government and its ability to handle child porn is directly deprived from a explicit legislative carveout for them by name. What they do would be a felony for you or I to do. The fact that they are technically non-profit rather than an agency makes them significantly less accountable to the public. We cannot FOIA their communications, their composition isn't subject to public review, we cannot vote them out. And we have no way to tell what their database contains, nor is there any avenue for redress should we somehow learn of an inappropriate listing.

To the extent that you can say that they're not exactly a government agency, they absolutely have been deputized by the government.