Policy makers in regions such as Europe are increasingly concerned about the trustworthiness and sovereignty of the foundations of their digital economy, because it often depends on systems operated or manufactured elsewhere. To help curb this problem, we propose the novel notion of a responsible Internet, which provides higher degrees of trust and sovereignty for critical service providers (e.g., power grids) and all kinds of other users by improving the transparency, accountability, and controllability of the Internet at the network-level. A responsible Internet accomplishes this through two new distributed and decentralized systems. The first is the Network Inspection Plane (NIP), which enables users to request measurement-based descriptions of the chains of network operators (e.g., ISPs and DNS and cloud providers) that handle their data flows or could potentially handle them, including the relationships between them and the properties of these operators. The second is the Network Control Plane (NCP), which allows users to specify how they expect the Internet infrastructure to handle their data (e.g., in terms of the security attributes that they expect chains of network operators to have) based on the insights they gained from the NIP. We discuss research directions and starting points to realize a responsible Internet by combining three currently largely disjoint research areas: large-scale measurements (for the NIP), open source-based programmable networks (for the NCP), and policy making (POL) based on the NIP and driving the NCP. We believe that a responsible Internet is the next stage in the evolution of the Internet and that the concept is useful for clean slate Internet systems as well.