Deep search agents have emerged as a promising paradigm for addressing complex information-seeking tasks, but their training remains challenging due to sparse rewards, weak credit assignment, and limited labeled data. Self-play offers a scalable route to reduce data dependence, but conventional self-play optimizes students only through sparse outcome rewards, leading to low learning efficiency. In this work, we observe that self-play naturally produces a question construction path (QCP) during task generation, an intermediate artifact that captures the reverse solution process. This reveals a new source of privileged information: self-play can provide high-quality privileged information for the self-distillation at low cost and at scale, without relying on human feedback or curated privileged information. Leveraging this insight, we propose Privileged Information Self-Play ($π$-Play), a novel multi-agent self-evolution framework combining self-play and self-distillation. In $π$-Play, an examiner generates tasks together with QCPs, and a teacher employs QCP as privileged context to densely supervise a student via self-distillation. This design transforms sparse-reward self-play into a dense-feedback co-evolution. Extensive experiments show that data-free $π$-Play surpasses fully supervised search agents and improves evolutionary efficiency by 2-3$\times$ over conventional self-play. Code is available at https://github.com/zhyaoch/pi-play.