Synthesizing missing modalities in multi-modal magnetic resonance imaging (MRI) is vital for ensuring diagnostic completeness, particularly when full acquisitions are infeasible due to time constraints, motion artifacts, and patient tolerance. Recent unified synthesis models have enabled flexible synthesis tasks by accommodating various input-output configurations. However, their training and evaluation are typically restricted to a single dataset, limiting their generalizability across diverse clinical datasets and impeding practical deployment. To address this limitation, we propose PMM-Synth, a personalized MRI synthesis framework that not only supports various synthesis tasks but also generalizes effectively across heterogeneous datasets. PMM-Synth is jointly trained on multiple multi-modal MRI datasets that differ in modality coverage, disease types, and intensity distributions. It achieves cross-dataset generalization through three core innovations: a Personalized Feature Modulation module that dynamically adapts feature representations based on dataset identifier to mitigate the impact of distributional shifts; a Modality-Consistent Batch Scheduler that facilitates stable and efficient batch training under inconsistent modality conditions; and a selective supervision loss to ensure effective learning when ground truth modalities are partially missing. Evaluated on four clinical multi-modal MRI datasets, PMM-Synth consistently outperforms state-of-the-art methods in both one-to-one and many-to-one synthesis tasks, achieving superior PSNR and SSIM scores. Qualitative results further demonstrate improved preservation of anatomical structures and pathological details. Additionally, downstream tumor segmentation and radiological reporting studies suggest that PMM-Synth holds potential for supporting reliable diagnosis under real-world modality-missing scenarios.