Hi Mari,
The concept of project ownership is different between Bitbucket Server and Bitbucket Cloud.
In Bitbucket Server, you can use the /rest/api/1.0/projects/{projectKey}/repos REST API (see this link for more details) to retrieve all the private repositories from a user. You'll need to replace {project} with the ~username (note the ~ as a prefix).
If you are interested to get the list of repositories that a user can access, you could check the instructions on the How to report on permissions by querying the database page.
Cheers, Caterina - Atlassian
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