Probably nothing that we haven't already heard. But Sen. Schumer and Sen. Salazar, and Sen. Specter (to the extent that he can be distinguished from them) want Dobson to let them in on the briefing.
The irony here is that conservatives are the ones that have been worried about Miers, while Harry Reid put her on the "approve" list he submitted to the White House. My guess is that Rove & Bush wanted to use Dobson to reassure social conservatives, since he's as likely as anyone to be a tough critic of this nomination. It wouldn't surprise me in the least if Dobson's little aside about "things I probably shouldn't know" is calculated more to enhance his reputation that Miers's.
While I don't really believe that any backroom deals have been cut, using Dobson as the evangelical pointman on this nomination probably wasn't the most savvy move available. In the past, his comments have gone a long way towards persuading me that he wants conservative activists, not Constitutionalists.
In any event, the notion of Senate Democrats, of all people, decrying the "demeaning of the process," would be funny if it weren't so relevant. While backroom deals to get on the Court would make a mockery of the process, the fact is Rove is in no position to make such promises on Miers's behalf, and Dobson would be powerless to enforce them, in any event.
Neither such set of condition obtains with regard to Senate Democrats, who only seem to be elected for life, but who in fact were caught taking marching orders from their own interest groups on judicial nominations.