I tried to create a Trello board just now and the default visibility setting is private. I assume that hasn't always been the case, otherwise people really went out of their way to make these boards public.
I don't think that was always the case. Recently I was googling my name and I found out I have bunch of Trello boards for various student projects that I completely forgot about (think "startups" but we've never gotten anywhere with them). I was surprised to see that. I don't think we deliberately made it public.