When they have more influence and impact on citizens than governments, perhaps we should revisit our definition of what "public" means. They can close businesses, make disappear the reach of specific competitors and do all sorts of things that governments cannot do because they agreed with their respective societies to limit their own power after thousands of years of history.
Right now the only part in which are they private is that they were founded by private capital.
It is utterly absurd to suggest that YouTube has more influence and impact than the government. The government has the power to imprison and murder you with impunity, their level of impact and influence completely dwarfs every video sharing site on the internet combined.
Right now the only part in which are they private is that they were founded by private capital.