A computer-implemented technique is described herein by which a first user can forward notification information to one or more other users, including a second user. The first user can interact with the notification information via a first computing device, while the second user can interact with the notification information using a second computing device. In some implementations, both the first computing device and the second computing device have access to an application which has generated the notification information or to which the notification information otherwise pertains. In other implementations, the second computing device may lack access to the application. In some implementations, the first user receives the notification information and then explicitly and manually controls the forwarding of the notification information (or a version of the notification information) to the second user. In other implementations, decision logic automatically controls the forwarding of the notification information to the second user.