Collaboration of LMs and SMs for domain tasksLarge language models support broad generalization but often require substantial data and computation for domain tasks. Small models are efficient and domain-specific, but they have limited general coverage. This survey studies collaboration between large and small models for private-domain adaptation. It focuses on cross-boundary settings where models and data are held by different parties, creating constraints on privacy, security, integrity, and resources. The paper organizes prior work by information flow: downward knowledge transfer from large models to small models, upward transfer from small models to large models, and inference-time collaboration across parties. It further identifies deployment challenges and frames practical collaboration as a multi-objective optimization problem.