You can include data from private Bitbucket Cloud projects as well as from Bitbucket Data Center instances. When including content from private repositories, you will get the following message in place of the content:
Once the page is published, the users can authenticate with Bitbucket by clicking on the Authenticate yourself link.
If the user is already authenticated, all content included from repositories the user has access to in Bitbucket will be visible.
You can find here a summary of how authentication is handled for both cases here.