The fixed spectrum assignment policy in the space sector and large constellations of low Earth orbit (LEO) satellites left little or no spectrum available for future LEO satellite communications services. Cognitive radio (CR) technology enables spectrum sharing between primary and secondary users without limiting the transmission power, and thus is of great interest to commercial and defense entities. A number of radio environment map (REM) techniques have been making cognitive radio networks (CRNs) practical by constructing a comprehensive map of the CRN by utilizing multi-domain information from geolocation databases, characteristics of spectrum use, geographical terrain models, propagation environment, and regulations. In this paper, we investigate spectrum sharing for a network comprised of a geostationary orbit (GEO) satellite and a LEO satellite with a multi-beam antenna array. A CRN architecture of GEO and LEO satellites shared downlink spectrum is proposed in various dimensions ranging from its architecture and REM structure to aggregation of channel utilization data as well as a frequency slot assignment mechanism.