There is increasing interest in developing personalized Task-oriented Dialogue Systems (TDSs). Previous work on personalized TDSs often assumes that complete user profiles are available for most or even all users. This is unrealistic because (1) not everyone is willing to expose their profiles due to privacy concerns; and (2) rich user profiles may involve a large number of attributes (e.g., gender, age, tastes, . . .). In this paper, we study personalized TDSs without assuming that user profiles are complete. We propose a Cooperative Memory Network (CoMemNN) that has a novel mechanism to gradually enrich user profiles as dialogues progress and to simultaneously improve response selection based on the enriched profiles. CoMemNN consists of two core modules: User Profile Enrichment (UPE) and Dialogue Response Selection (DRS). The former enriches incomplete user profiles by utilizing collaborative information from neighbor users as well as current dialogues. The latter uses the enriched profiles to update the current user query so as to encode more useful information, based on which a personalized response to a user request is selected. We conduct extensive experiments on the personalized bAbI dialogue benchmark datasets. We find that CoMemNN is able to enrich user profiles effectively, which results in an improvement of 3.06% in terms of response selection accuracy compared to state-of-the-art methods. We also test the robustness of CoMemNN against incompleteness of user profiles by randomly discarding attribute values from user profiles. Even when discarding 50% of the attribute values, CoMemNN is able to match the performance of the best performing baseline without discarding user profiles, showing the robustness of CoMemNN.