I realize that "influence" can be a dirty word, in some contexts, but remember that our participatory government was designed to be influenced. It's our government. We get a say, and part of the way we do that is by supporting candidates, often with money, who will go about their public duties in the ways we wish.
These doyens of government transparency and accountability are afraid that making recordings of their meetings and their email addresses available online, where the press and the public can easily access them, might put them at risk of identity theft. I'm not joking.
The people elected to the executive branch, or the Legislature, make the laws and execute the laws. Thus, the voters have a fundamentally different relationship to them than they do to the judicial branch.