Oil spills in the ocean pose severe environmental risks, making early detection essential. Synthetic aperture radar (SAR) based oil spill segmentation offers robust monitoring under various conditions but faces challenges due to the limited labeled data and inherent speckle noise in SAR imagery. To address these issues, we propose (i) a diffusion-based Data Augmentation and Knowledge Distillation (DAKD) pipeline and (ii) a novel SAR oil spill segmentation network, called SAROSS-Net. In our DAKD pipeline, we present a diffusion-based SAR-JointNet that learns to generate realistic SAR images and their labels for segmentation, by effectively modeling joint distribution with balancing two modalities. The DAKD pipeline augments the training dataset and distills knowledge from SAR-JointNet by utilizing generated soft labels (pixel-wise probability maps) to supervise our SAROSS-Net. The SAROSS-Net is designed to selectively transfer high-frequency features from noisy SAR images, by employing novel Context-Aware Feature Transfer blocks along skip connections. We demonstrate our SAR-JointNet can generate realistic SAR images and well-aligned segmentation labels, providing the augmented data to train SAROSS-Net with enhanced generalizability. Our SAROSS-Net trained with the DAKD pipeline significantly outperforms existing SAR oil spill segmentation methods with large margins.