Early detection of security bug reports (SBRs) is critical for timely vulnerability mitigation. We present an evaluation of prompt engineering and fine-tuning approaches for predicting SBRs using Large Language Models (LLMs). Our results revealed a distinct trade-off between the two approaches. Using the prompted proprietary models, we observed the highest sensitivity to SBRs with an average G-measure of 0.77 among all the tested datasets, albeit at the cost of a higher false-positive rate, resulting in a precision of only 0.22. In contrast, fine-tuned models offered more viable solutions, achieving an average G-measure of 0.51 with a 10 to 50 times reduction in inference latency compared to proprietary LLMs. While proprietary models offer accessibility, specialized fine-tuned models remain the optimal choice for privacy-preserving and cost-effective SBR prediction.