We actually do have one, but we’re not totally sold on documenting/supporting it as part of the platform without a fuller idea of use cases. DMs are a slightly curious piece of the Twitter platform at the best of times (and in the unauthenticated environment on Web Intents, it would be difficult to provide cues on whether a user can/can’t DM another user), however a basic interface to send them with fallback to public @mentions is possible.
Personally I’m game for it (which—sadly—isn’t a promise) but would love to hear your thoughts on how you’d use it. What kind of interactions do you want to put in your site/app that we should support here?
Ben