Large language model (LLM)-based multi-agent systems (MAS) improve reasoning through specialized collaboration, but dense multi-round communication can increase prompt cost and expose agents to outdated, redundant, or noisy context. A prominent line of MAS communication-efficiency work reduces communication through agent-, edge-, or topology-level structural decisions, which operate on graph structure rather than the realized text entering a specific receiver prompt. Such structural decisions are too coarse for runtime communication: messages sent over the same edge may carry useful evidence in one query, redundant content in another, or only a few useful spans. We introduce MessageSieve, a training-free runtime message pruning policy for fixed-graph MAS that operates on message instances rather than agents or edges. Before prompt assembly, it uses lightweight state cards and local message comparisons to preserve original text, retain compact evidence, or omit messages, without a trained selector or additional LLM call. We evaluate MessageSieve on two backbone models and five benchmarks covering expert QA, math reasoning, and code generation. Experiments show that MessageSieve improves the macro-average score over dense Full MAS by 4.10 points on Qwen3-8B and 2.59 points on Gemini 3.1 Flash-Lite, while reducing evaluation tokens by 19–49%.