When you control for benefits and job stability, I think people in the public sector are better compensated than in the private sector on average. Biggs and Richwine found this to be the case for public school teachers.
It's pretty silly to look at teachers, because the markets are highly unequal. The public sector marketplace for teaching is vast compared to the private sector, and it's also far more demanding, with greater requirements for credentialing, etc.
Where the marketplace is robust in both sectors, you tend to see competitive pricing between the two. I've been an IT worker in both private and public sectors, which is in demand everywhere, and I found my wage/benefits structures in both to be comparable. Public sector features like comp time are offset by private sector features like annual bonuses. Other ubiquitous jobs get the same treatment.
And where the private marketplace is almost nonexistent, workers are screwed. In most areas of this country, we pay a pittance to social workers, law enforcement, etc. It's a wonder anyone takes those jobs.