Publish finished build to another Jenkins
The Build Publisher Plugin reads:
[...] then send the results to another Jenkins which is facing the outside world.
[...] build records from private Jenkins is sent to public Jenkins, through HTTP.
I interpret this as the results being published just for being viewed on the public server.
Just like a book publisher: He or she doesn't write books or continuations of books. He or she just publishes them for reading.
See Calling a jenkins build from outside of jenkins? for how to trigger a build on a Jenkins server from remote, i.e. from your private on your public.