In this study, we present an approach to train a single speech enhancement network that can perform both personalized and non-personalized speech enhancement. This is achieved by incorporating a frame-wise conditioning input that specifies the type of enhancement output. To improve the quality of the enhanced output and mitigate oversuppression, we experiment with re-weighting frames by the presence or absence of speech activity and applying augmentations to speaker embeddings. By training under a multi-task learning setting, we empirically show that the proposed unified model obtains promising results on both personalized and non-personalized speech enhancement benchmarks and reaches similar performance to models that are trained specialized for either task. The strong performance of the proposed method demonstrates that the unified model is a more economical alternative compared to keeping separate task-specific models during inference.