The ultra-dense deployment of interconnected satellites will characterize future low Earth orbit (LEO) mega-constellations. Exploiting this towards a more efficient satellite network (SatNet), this paper proposes a novel LEO SatNet architecture based on distributed massive multiple-input multiple-output (DM-MIMO) technology allowing ground user terminals to be connected to a cluster of satellites. To this end, we investigate various aspects of DM-MIMO-based satellite network design, the benefits of using this architecture, the associated challenges, and the potential solutions. In addition, we propose a distributed joint power allocation and handover management (D-JPAHM) technique that jointly optimizes the power allocation and handover management processes in a cross-layer manner. This framework aims to maximize the network throughput and minimize the handover rate while considering the quality-of-service (QoS) demands of user terminals and the power capabilities of the satellites. Moreover, we devise an artificial intelligence (AI)-based solution to efficiently implement the proposed D-JPAHM framework in a manner suitable for real-time operation and the dynamic SatNet environment. To the best of our knowledge, this is the first work to introduce and study DM-MIMO technology in LEO SatNets. Extensive simulation results reveal the superiority of the proposed architecture and solutions compared to conventional approaches in the literature.