Regarding on what error code should be used if incorrect/insufficient private claim is used. Should it be 401 (Unauthorized), or 403 (Forbidden)?
Maybe this should be clarified in the specification.
3.1. Error Codes
When a request fails, the resource server responds using the
appropriate HTTP status code (typically, 400, 401, 403, or 405) and
includes one of the following error codes in the response:
invalid_request
The request is missing a required parameter, includes an
unsupported parameter or parameter value, repeats the same
parameter, uses more than one method for including an access
token, or is otherwise malformed. The resource server SHOULD
respond with the HTTP 400 (Bad Request) status code.
invalid_token
The access token provided is expired, revoked, malformed, or
invalid for other reasons. The resource SHOULD respond with
the HTTP 401 (Unauthorized) status code. The client MAY
request a new access token and retry the protected resource
request.
insufficient_scope
The request requires higher privileges than provided by the
access token. The resource server SHOULD respond with the HTTP
403 (Forbidden) status code and MAY include the "scope"
attribute with the scope necessary to access the protected
resource.
If the request lacks any authentication information (e.g., the client
was unaware that authentication is necessary or attempted using an
unsupported authentication method), the resource server SHOULD NOT
include an error code or other error information.
For example:
HTTP/1.1 401 Unauthorized
WWW-Authenticate: Bearer realm="example"