You're right that it doesn't make sense to compare the groups at an aggregate level but look at
http://www.usatoday.com/news/nation/2010-03-04-federal-pay_N... for an attempt to look at comparable job catagories.
For example, registered nurses working for the government received $74,460 per year while in the private sector, the number is $63,780.
The same study found some occupations where the private sector paid more (airline pilots, lawyers).
There is a nice table there with a breakdown by occupation.
So it seems like the issue is a bit more complicated than the headlines would suggest. Imagine that.