We present a novel method that estimates confidence map of an initial disparity by making full use of tri-modal input, including matching cost, disparity, and color image through deep networks. The proposed network, termed as Locally Adaptive Fusion Networks (LAF-Net), learns locally-varying attention and scale maps to fuse the tri-modal confidence features. The attention inference networks encode the importance of tri-modal confidence features and then concatenate them using the attention maps in an adaptive and dynamic fashion. This enables us to make an optimal fusion of the heterogeneous features, compared to a simple concatenation technique that is commonly used in conventional approaches. In addition, to encode the confidence features with locally-varying receptive fields, the scale inference networks learn the scale map and warp the fused confidence features through convolutional spatial transformer networks. Finally, the confidence map is progressively estimated in the recursive refinement networks to enforce a spatial context and local consistency. Experimental results show that this model outperforms the state-of-the-art methods on various benchmarks.