Massive machine-type communication (MTC) addresses new IoT use cases such as connected vehicles, smart agriculture, and factory automation. However, major concerns face massive MTC applications, such as data privacy, latency constraints, and communication overhead. Distributed learning, an emerging technology that enables edge users to train various machine learning (ML) models without sharing raw data, is very promising for overcoming such concerns. However, distributed learning still faces some difficulties related to the amount of training data, accuracy, complexity, and dynamic wireless environment. Herein, we elucidate different distributed learning methodologies, their limitations, and case studies in the spirit of massive MTC. Specifically, the paper showcases the pros of Federated Learning (FL) in the context of dense MTCbased handwritten digits classification. After that, we highlight the differences between Fl and Federated Distillation (FD) and proceed to split, transfer, and meta-learning. Then, we discuss multi-agent reinforcement learning (MARL) in a UAV swarm empowered smart agriculture scenario. Finally, we present some challenges and future directions for distributed learning research in the context of massive MTC.