Bug severity prediction is a critical task in software engineering as it enables more efficient resource allocation and prioritization in software maintenance. While AI-based analyses and models significantly require access to extensive datasets, industrial applications face challenges due to data-sharing constraints and the limited availability of labeled data. In this study, we investigate method-level bug severity prediction using source code metrics and Large Language Models (LLMs) with two widely used datasets. We compare the performance of models trained using centralized learning, federated learning, and synthetic data generation. Our experimental results, obtained using two widely recognized software defect datasets, indicate that models trained with federated learning and synthetic data achieve comparable results to centrally trained models without data sharing. Our finding highlights the potential of privacy-preserving approaches such as federated learning and synthetic data generation to enable effective bug severity prediction in industrial context where data sharing is a major challenge. The source code and dataset are available at our GitHub repository: https://github.com/drvshavva/EASE2025-Privacy-Preserving-Methods-for-Bug-Severity-Prediction.