You can have access to the datasets that were used to create a Tableau Public dashboard if you download it.
Tableau Desktop dashboards, I believe, are the same.
Which leads me to the question: can anyone with access to a Tableau Server download Tableau Desktop documents?
I'd like to publish a Tableau Desktop dashboard to Tableau Server so that I can embed it on a website, but I don't want visitors to be able to download it. Knowing this will most likely influence my decision to purchase Tableau Server.