Visible thermal person re-identification (VT-ReID) is a cross-modality pedestrian retrieval problem, which automatically searches persons between day-time visible images and night-time thermal images. Despite the extensive progress in single-modality ReID, the cross-modality pedestrian retrieval problem has limited attention due to its challenges in modality discrepancy and large intra-class variations across cameras. Existing cross-modality ReID methods usually solve this problem by learning cross-modality feature representations with modality-sharable classifier. However, this learning strategy may lose discriminative information in different modalities. In this paper, we propose a novel modality-aware collaborative (MAC) learning method on top of a two-stream network for VT-ReID, which handles the modality-discrepancy in both feature level and classifier level. In feature level, it handles the modality discrepancy by a two-stream network with different parameters. In classifier level, it contains two separate modality-specific identity classifiers for two modalities to capture the modality-specific information, and they have the same network architecture but different parameters. In addition, we introduce a collaborative learning scheme, which regularizes the modality-sharable and modality-specific identity classifiers by utilizing the relationship between different classifiers. Extensive experiments on two cross-modality person re-identification datasets demonstrate the superiority of the proposed method, achieving much better performance than the state-of-the-art.