It is not mentioned in the API docs that you can not send a private message to somebody who does not follow you. (https://dev.twitter.com/docs/api/1.1/post/direct_messages/new)
Also this error code is not listed on https://dev.twitter.com/docs/error-codes-responses.
Is there any possibility to send a private message so somebody who does not follow me? (E.g. after he mentioned me in one of his tweets)
Thanks for the feedback, good points, we will work on updating the documentation!
Currently there is no way to send a message (DM) to someone who does not follow a user.
Yes, there is. User whom you want to send a DM and isn’t in your followers list, has to have a DM option active. (Go to Security and privacy settings page and mark DM option). Regards.
Thanks - my post was correct in July 2014, but since then the option to have others DM you, if you enable the option to let anyone DM you in settings.