How to have a private repository with public issue tracker on GitHub?

Not possible as of 2019-09 as documented at: https://help.github.com/articles/issues-only-access-permissions/ (archive):

GitHub does not provide issues-only access permissions, but you can accomplish this using a second repository which contains only the issues.

They suggest that you create a separate public repository just for the issues instead.

Possible same request: Github-like hosting, with private source, public ticketing system

Bitbucket provides this. You can keep the code private but make tracker public (anyone can view, create, and comment on issues).