I'm playing a bit more with this and I figured out the problem with permissions might be broader.
I added: logging.dest: /var/log/kibana.log to kibana.yml. That prevents kibana from starting. I fails with a permition error: Error: EACCES: permission denied, open '/var/log/kibana.log'
In case it is expected behavior that the log file should be owned by a kibana user it should state so in the comments. This also makes me wonder if any other configs are subjected to the same problems (pid files, certificates etc.).
I tried to run it "manually" with sudo bin/kibana and the permissions problems (reporting plugin, custom log file) went away. So I came to the conclusion that sudo service kibana start just runs it without the necessary permissions to actually work.
I've installed it as a deb package on a Ubuntu 16 OS.