Multi-sensory systems for embodied intelligence, from wearable body-sensor networks to instrumented robotic platforms, routinely face a sensor-asymmetry problem: the richest modality available during laboratory data collection is absent or impractical at deployment time due to cost, fragility, or interference with physical interaction. We introduce PULSE, a general framework for privileged knowledge transfer from an information-rich teacher sensor to a set of cheaper, deployment-ready student sensors. Each student encoder produces shared (modality-invariant) and private (modality-specific) embeddings; the shared subspace is aligned across modalities and then matched to representations of a frozen teacher via multi-layer hidden-state and pooled-embedding distillation. Private embeddings preserve modality-specific structure needed for self-supervised reconstruction, which we show is critical to prevent representational collapse. We instantiate PULSE on the wearable stress-monitoring task, using electrodermal activity (EDA) as the privileged teacher and ECG, BVP, accelerometry, and temperature as students. On the WESAD benchmark under leave-one-subject-out evaluation, PULSE achieves 0.994 AUROC and 0.988 AUPRC (0.965/0.955 on STRESS) without EDA at inference, exceeding all no-EDA baselines and matching the performance of a full-sensor model that retains EDA at test time. We further demonstrate modality-agnostic transfer with ECG as teacher, provide extensive ablations on hidden-state matching depth, shared-private capacity, hinge-loss margin, fusion strategy, and modality dropout, and discuss how the framework generalizes to broader embodied sensing scenarios involving tactile, inertial, and bioelectrical modalities.