We propose a federated learning-like framework, Federation over Text (FoT), that enables multiple clients solving different tasks to collectively generate a shared library of metacognitive insights by iteratively federating their local reasoning processes without sharing actual problem instances or task instructions. Instead of federation over gradients (e.g., as in distributed training), FoT operates at the semantic level without any gradient optimization or supervision signal. Iteratively, each client runs an LLM agent that does local thinking and self-improvement on their specific tasks independently, and shares reasoning traces with a central server, which aggregates and distills them into a cross-task (and cross-domain) insight library that existing and future agents can leverage to improve performance on related tasks. Experiments show that FoT improves reasoning effectiveness and efficiency across a wide range of challenging applications, including mathematical problem solving, cross-domain collaboration, real-world daily tasks, and machine learning research insight discovery. Specifically, it improves average performance scores by 25% while reducing the reasoning tokens by 4% across the first three applications. In the research insight discovery application, FoT is able to generate insights that cover over 80% of the major contributions in the subsequent papers.