If I'm not mistaken, there's currently no way to determine whether the authenticated user has write access to a specific repository using the Bitbucket API.
I know it is possible to determine whether it has "admin" access using the "privileges" resource. Similar goes for "read" access. Nonetheless determining if the user has "write" access is not possible without trying to actually write to the repository.
But determining "write"/"admin" access in advance is crucial for any serious application built upon the API.
My proposal is simply adapting following resource: https://confluence.atlassian.com/display/BITBUCKET/privileges+Endpoint#privilegesEndpoint-GETprivilegesforanindividual It is currently restricted to: "only the repository owner, a team account administrator, or an account with administrative rights on the repository can make this call."
Making this resource available to the user that is being queried (so the authenticated user can query this resource about himself for every repository) would solve the whole problematic, providing a clean way of determining the access level to a repository.