I really like it when my specialist has a prominent public profile.
A recent case: tenure at a major university, had written or was quoted in many good journals and papers, and had a number of impressive accreditations (for example he was also head of his specialty's medical association for the country). This showed dedication and interest as well as some measure of intelligence.
I was accordingly confident he was on top of the latest developments in his field and would be able to deal with complications better than your run of the mill specialist. The care was great throughout.
Similarly, our lawyer has been quoted twice by the Supreme Court in Singapore, has written many articles in the Straits Times, and does great (and visible) pro bono work, all of which I could find off a quick Google for his name. Whilst I hired him based on a recommendation, it was comforting to see this public confirmation of his ability and interest in the field.