plugin installation via Admin Dashboard (GreenHopper, Bonfire) doesn't work on a fresh JIRA installation

Sample error message:

"GreenHopper - Agile project management for JIRA install failed. Status: 401. URL: http://127.0.0.1:8080/rest/plugins/1.0/?token=5478999153568121197"

The reason (as identified by edalgliesh) is lack of administration permissions, and so the expected behaviour is for a web sudo dialog to show up beforehands.