Breast cancer is becoming a significant global health challenge, with millions of fatalities annually. Magnetic Resonance Imaging (MRI) can provide various sequences for characterizing tumor morphology and internal patterns, and becomes an effective tool for detection and diagnosis of breast tumors. However, previous deep-learning based tumor segmentation methods from multi-parametric MRI still have limitations in exploring inter-modality information and focusing task-informative modality/modalities. To address these shortcomings, we propose a Modality-Specific Information Disentanglement (MoSID) framework to extract both inter- and intra-modality attention maps as prior knowledge for guiding tumor segmentation. Specifically, by disentangling modality-specific information, the MoSID framework provides complementary clues for the segmentation task, by generating modality-specific attention maps to guide modality selection and inter-modality evaluation. Our experiments on two 3D breast datasets and one 2D prostate dataset demonstrate that the MoSID framework outperforms other state-of-the-art multi-modality segmentation methods, even in the cases of missing modalities. Based on the segmented lesions, we further train a classifier to predict the patients' response to radiotherapy. The prediction accuracy is comparable to the case of using manually-segmented tumors for treatment outcome prediction, indicating the robustness and effectiveness of the proposed segmentation method. The code is available at https://github.com/Qianqian-Chen/MoSID.