Large Language Models (LLMs) like ChatGPT, Google Bard, Claude, and Llama 2 have revolutionized natural language processing and search engine dynamics. However, these models incur exceptionally high computational costs. For instance, GPT-3 consists of 175 billion parameters and inference on these models also demands billions of floating-point operations. Caching is a natural solution to reduce LLM inference costs on repeated queries. However, existing caching methods are incapable of finding semantic similarities among LLM queries, leading to unacceptable false hit-and-miss rates. This paper introduces MeanCache, a semantic cache for LLMs that identifies semantically similar queries to determine cache hit or miss. Using MeanCache, the response to a user's semantically similar query can be retrieved from a local cache rather than re-querying the LLM, thus reducing costs, service provider load, and environmental impact. MeanCache leverages Federated Learning (FL) to collaboratively train a query similarity model in a distributed manner across numerous users without violating privacy. By placing a local cache in each user's device and using FL, MeanCache reduces the latency and costs and enhances model performance, resulting in lower cache false hit rates. Our experiments, benchmarked against the GPTCache, reveal that MeanCache attains an approximately 17% higher F-score and a 20% increase in precision during semantic cache hit-and-miss decisions. Furthermore, MeanCache reduces the storage requirement by 83% and accelerates semantic cache hit-and-miss decisions by 11%, while still surpassing GPTCache.