Is it possible to include a private repository?

Yes, a private repository can be included from Bitbucket Cloud. Just paste link of file from Bitbucket Cloud private repo, and the app will show a link to grant access to the repository.

To view a private repository in Confluence Cloud, does the user also have to be a Bitbucket Cloud user?

Yes, the Confluence Cloud user also has to be a Bitbucket Cloud user and member of the team where the repository is stored.

Can we Impersonate a Bitbucket user, so Confluence Users do not need to Login to Bitbucket

It depends on where the file is, on Bitbucket Cloud or Bitbucket Data Center:

Bitbucket Edition	Authentication
Bitbucket Cloud	Each Confluence user needs to authenticate against Bitbucket Cloud. So every Confluence user also needs a Bitbucket valid login. The user needs permissions on Bitbucket to see the file.
Bitbucket Data Center	A personal access token is used for all Confluence Users. A Confluence user does not need any Bitbucket login. Every Confluence user can view included Bitbucket files.

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