Quite a bit, apparently, according to Sunlight Foundation Intern Alyssa Noronha:
I discovered that increased press freedom may help foster a growth in e-government, and growth of these two measures greatly impacts transparency.
Ms. Noronha also compiled data from Reporters Without Borders, the Brookings Institute, and the CIA Factbook and found that
[T]here is a statistically significant relationship between press freedom and e-government, holding the size of the economy (GDP) constant. As press freedom increases by ten points on a 100-point scale, e-government measures increase by approximately one point (.8) on a 100-point scale, holding GDP constant.
Beyond that, transparency also takes root in countries with strong traditions of free press and open and electronic government.
A free press supports accountability by writing about corruption in government and demanding more information in more accessible forms. By putting data and information in the hands of citizens, who can then analyze and interpret the data themselves, you can break down a lot of barriers to participation. Increased press freedom demands more transparency, and aims to create a more responsive, accountable government.
Congratulations to Ms. Noronha on her excellent work.