You need to authorize the token for the specific user that will receive the status update. One option is to follow the PIN-based OAuth method for sending updates to another user's status.
From the Implementation section:
The PIN-based flow is implemented in the same exact way as
Implementing Sign in with Twitter and 3-legged Authorization, with
the only difference being that the value for oauth_callback must be
set to oob during the POST oauth / request_token call.
Once your application has obtained either a GET oauth/authenticate or
GET oauth/authorize URL, display the URL to the user so that they may
use a web browser to access Twitter.
When an oob callback is requested and the user visits Twitter, they
will not be automatically redirected to the application upon approving
access. Instead, they will see a PIN code, with instructions to return
to the application and enter this value.
Your application must allow the user to input this PIN to complete the
flow. The PIN code must be passed as the value for oauth_verifier for
the POST oauth/access_token request. All other requests will work
normally.