A computer-implemented technique is described herein for presenting notifications on a user interface presentation of a recipient computing device. The technique is considered tiered because it distinguishes between first-type (people-centric) notifications and second-type notifications, and because it processes and presents first-type notifications in a privileged manner compared to the second-type notifications. For instance, the technique presents content associated with each first-type notification in an allocated window that appears in positional association with a person indicator, where that person indicator identifies the person who sent the first-type notification; the technique presents each second-type notification in a different manner. In some implementations, an application developer who designs a custom first-type notification is given authority to draw within the allocated window that is used to present the custom first-type notification in a manner specified by the application developer.