Many real-world applications require language models to internalize knowledge that was absent from pretraining, to incorporate continually updated facts, or to learn new domain skills. Self-distillation has emerged as a natural approach for this task. A model conditioned on the source material acts as the teacher, and an unconditioned student is trained to reproduce the teacher's behavior closed-book. We propose Speculative Self-Distillation (SSD), an efficient per-token mixed-policy method for self-distillation. The student generates each rollout by default, while the teacher intervenes only at positions where its next-token distribution substantially diverges from the student's. Unlike off-policy distillation with teacher-generated prefixes, this design keeps training rollouts close to the student's test-time behavior, while avoiding the low-signal states that fully on-policy rollouts drift into. In effect, SSD lets the teacher steer the rollout exactly where the privileged context changes the prediction. On three closed-book QA benchmarks measuring fact recall, knowledge updates, and skill acquisition, SSD nearly matches on-policy distillation while using 38-69% fewer supervised tokens and consistently outperforms off-policy distillation on the held-out test set. On a forgetting stress test, SSD retains the most general capability while also achieving the best in-domain accuracy and using less supervision than the on-policy baseline. Overall, SSD establishes a new accuracy-efficiency-forgetting Pareto frontier for self-distillation.

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