Twitter says that the content (tweets) belong to you, while the service reserves the right to distribute it freely, including via API applications.
Twitter Support :: The Twitter Rules
The Twitter Rules
Our goal is to provide a service that allows you to discover and receive content from sources that interest you as well as to share your content with others. We respect the ownership of the content that users share and each user is responsible for the content he or she provides. Because of these principles, we do not actively monitor user’s content and will not censor user content, except in limited circumstances described below.