The joint optimization of image-based (I2I) and text-based (T2I) person re-identification (ReID) is hindered by modality discrepancies and conflicting training objectives, leading to suboptimal shared representations. While I2I ReID focuses on identity-level invariance across images of the same person, T2I ReID is driven by instance-specific textual descriptions tied to unique visual traits. This paper explores the fundamental difference between two ReID tasks and their optimization processes for effective training. Since I2I and T2I ReID are often studied separately, the loss functions optimized for one retrieval setting may negatively affect the representation quality required by the other. Motivated by these findings, we propose a decoupled two-stage training pipeline for learning a shared representation across image and text modalities. The pipeline is based on a single vision encoder that supports both I2I and T2I retrieval while avoiding cross-task interference during training. We provide extensive experiments across multiple configurations, varying domain mixing procedures, learning strategies, and task objectives. We observed that I2I ReID pre-training positively impacts the generalization ability to T2I data. Besides, we find that incorporating textual supervision during the vision encoder training stage enhances both I2I and T2I performance. We believe our insights provide a meaningful step toward unified ReID systems and cross-modal retrieval overall.