Unlike on-policy distillation (OPD), which depends on a separate teacher model, on-policy self-distillation (OPSD) leverages a single language model acting simultaneously as the student and the teacher, with the teacher conditioned on privileged information. Despite this architectural simplicity, synchronous rollout generation dominates overall training time, creating a severe systems bottleneck when scaling OPSD. To address this, we introduce Asynchronous On-Policy Self-Distillation (AsyncOPSD), a framework that decouples rollout generation from learner updates to improve training throughput. Since the teacher and student share parameters, AsyncOPSD eliminates the multi-model coordination overhead inherent to OPD architectures in asynchronous training. As the student policy evolves, buffered trajectories generated by older policies become stale, potentially degrading performance. To mitigate this, we introduce a distillation objective based on staleness-aware policy control and token-level entropy weighting. Across three model sizes and mathematical reasoning benchmarks, AsyncOPSD improves the training throughput by $1.96$--$2.25\times$ over synchronous OPSD while increasing average benchmark accuracy.

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