So you would rather trust the government instead? They mostly work with private companies to run their IT stuff so at the end of the day your data would end up on some private servers anyway.
That's a false dichotomy - in many countries, there are public institutions that are (somewhat) independent on the government. (In Czech Republic, we have universities, radio and TV station that are run like that.)
It could also be a cooperative, or a non-profit.. there are many models.
I think the problem is slightly more complex for a large-scale social network. When we look at the cost of running something like facebook, we're looking at hiring an army of expensive developers (someone has to build that platform and maintain it), buying or renting an army of expensive servers to make sure it's up and running 24/7, designers so people can actually use it, extra offices in order to host everyone, etc. It seems to be a money issue as well as an organizational issue.
The scenario we'd be looking at would be a small internal network for a university or a small organization. I believe any government could handle such a thing. For something like facebook we're looking at a lot of money and a bunch of skilled people. Not sure that's in line with governments...
It could also be a cooperative, or a non-profit.. there are many models.