During any given Kibana browser session, the data requests are forwarded to Elasticsearch, some other calls are done by the Kibana server process.
The username and password you set in kibana.yml as per our plugin installation instructions, will be used by the kibana server process. The browser will need to provide their own basic auth credentials.
So to answer your question, a user will log in using their browser’s default basic auth prompt when browsing Kibana (without our non-free Kibana plugin). It’s going to be a less than smooth experience because there won’t be a logout button and some Kibana apps will show errors.