We focus on characterizing common and different coexpression patterns among RNAs and proteins in breast cancer tumors. To address this problem, we introduce Joint Random Forest (JRF), a novel nonparametric algorithm to simultaneously estimate multiple coexpression networks by effectively borrowing information across protein and gene expression data. The performance of JRF was evaluated through extensive simulation studies using different network topologies and data distribution functions. Advantages of JRF over other algorithms that estimate class-specific networks separately were observed across all simulation settings. JRF also outperformed a competing method based on Gaussian graphic models. We then applied JRF to simultaneously construct gene and protein coexpression networks based on protein and RNAseq data from CPTAC-TCGA breast cancer study. We identified interesting common and differential coexpression patterns among genes and proteins. This information can help to cast light on the potential disease mechanisms of breast cancer.