If you download a Tableau Public dashboard, you'll get access to the datasets that where use to make it.
I believe it is the same for Tableau Desktop dashboards.
Which leads me to : are Tableau Desktop documents, stored on a Tableau Server, downloadable by anyone with access to that link ?
I would like to publish a Tableau Desktop dashboard on a Tableau Server so I can put it on a website yet I don't want the viewers to be able to download the dashboard. Knowing this will likely determine whether or not I buy Tableau Server.
Short answer is yes, people who have access to download (which you set in the user or group permissions) will be able to see whatever data it is connected to. If it is connected to a database such as SQL the user would be prompted to enter credentials that are allowed to connect to that database. If the data is in the form of an excel document then it would just be available with the download.
It sounds to me all you really need is to just limit the users from having download access, which is changed in the user/group settings. Just let them "view only".
Hope that helps!
To add to the answer above:
Not only in Tableau Server can you restrict users from downloading your data, also in Tableau Public. Under 'Edit details' of your dashboard, you will find a checkbox for "Allow workbook and its data to be downloaded by others". If you uncheck this, viewers will not be able to download the workbook or it's underlying data.
To add to your real intentions with this question: As far as I understand your question, you want to use Tableau Server to present you data on a website. Is this website meant for everyone to be accessible without signing in to Tableau Server? If so, then this will not be possible with a regular Tableau Server license, only with the (very expensive) core license. See the explanation on Tableau's website about embedded views: Embedded views
To connect to Tableau Online, select Tableau Online under Quick Connect.
Tip: If you don't sign out, Tableau Desktop saves your server connection, so you can skip step 3. You can also skip it if Kerberos is enabled on Tableau Server and your computer has valid credentials. For more information, see Automatically keep Tableau Desktop connected to Tableau Server or Online.
Select a data source from the list of published data sources. (Data that you don't have Connect permissions for is grayed out.)
Note: If you select a cube (multidimensional) data source, the Create Local Copy dialog box appears, and you must create a local copy of the data before you can start your analysis.
Search for data by using the Search box. Or you can sort the list by selecting the column headers. Column headers that support sorting are underlined when you hover over the header. You can also select the filter icon to filter data by Connection Type and by Certified.
When Tableau Catalog is enabled,
Note: From the Search for Data results, you can make a connection with a subset of the data connectors that Tableau Desktop supports. If a data source, database, file, or table is grayed out, you can't connect from Tableau Server or Tableau Online. You can, however, connect from the Tableau Desktop Connect pane, if you have the correct permissions.
Data sources published to Tableau Server or Tableau Online are shown in the Data pane with a Tableau icon.
You might want to download a local copy of the data source so that you can, for example, work offline or make changes to a data source without modifying the original. To download a local copy, on the Data menu, select the data source, and then select Create Local Copy. A duplicate of the data source is added to the Data pane.