API does not allow posting anonymous issues
The API allows posting issues but only if authenticated with a valid account if the repository is a private one, even when the issue tracker is public: https://confluence.atlassian.com/display/BITBUCKET/issues+Resource
It doesn't make sense to have a feature missing from the API that is available through the ordinary browser. It would be very useful to allow anonymous issue posting this way, since it would enable the development of apps that have an internal bug reporting feature.
(however, anonymous issue posting in browser also appears broken at the time of writing - see issue