In medical imaging, inter-observer variability among radiologists often introduces label uncertainty, particularly in modalities where visual interpretation is subjective. Lung ultrasound (LUS) is a prime example-it frequently presents a mixture of highly ambiguous regions and clearly discernible structures, making consistent annotation challenging even for experienced clinicians. In this work, we introduce a novel approach to both labeling and training AI models using expert-supplied, per-pixel confidence values. Rather than treating annotations as absolute ground truth, we design a data annotation protocol that captures the confidence that radiologists have in each labeled region, modeling the inherent aleatoric uncertainty present in real-world clinical data. We demonstrate that incorporating these confidence values during training leads to improved segmentation performance. More importantly, we show that this enhanced segmentation quality translates into better performance on downstream clinically-critical tasks-specifically, estimating S/F oxygenation ratio values, classifying S/F ratio change, and predicting 30-day patient readmission. While we empirically evaluate many methods for exposing the uncertainty to the learning model, we find that a simple approach that trains a model on binarized labels obtained with a (60%) confidence threshold works well. Importantly, high thresholds work far better than a naive approach of a 50% threshold, indicating that training on very confident pixels is far more effective. Our study systematically investigates the impact of training with varying confidence thresholds, comparing not only segmentation metrics but also downstream clinical outcomes. These results suggest that label confidence is a valuable signal that, when properly leveraged, can significantly enhance the reliability and clinical utility of AI in medical imaging.