Big Artificial Intelligence (AI) models have emerged as a crucial element in various intelligent applications at the edge, such as voice assistants in smart homes and autonomous robotics in smart factories. Training big AI models, e.g., for personalized fine-tuning and continual model refinement, poses significant challenges to edge devices due to the inherent conflict between limited computing resources and intensive workload associated with training. Despite the constraints of on-device training, traditional approaches usually resort to aggregating training data and sending it to a remote cloud for centralized training. Nevertheless, this approach is neither sustainable, which strains long-range backhaul transmission and energy-consuming datacenters, nor safely private, which shares users' raw data with remote infrastructures. To address these challenges, we alternatively observe that prevalent edge environments usually contain a diverse collection of trusted edge devices with untapped idle resources, which can be leveraged for edge training acceleration. Motivated by this, in this article, we propose collaborative edge training, a novel training mechanism that orchestrates a group of trusted edge devices as a resource pool for expedited, sustainable big AI model training at the edge. As an initial step, we present a comprehensive framework for building collaborative edge training systems and analyze in-depth its merits and sustainable scheduling choices following its workflow. To further investigate the impact of its parallelism design, we empirically study a case of four typical parallelisms from the perspective of energy demand with realistic testbeds. Finally, we discuss open challenges for sustainable collaborative edge training to point to future directions of edge-centric big AI model training.