In this work, we propose a novel Convolutional Neural Network (CNN) architecture for the joint detection and matching of feature points in images acquired by different sensors using a single forward pass. The resulting feature detector is tightly coupled with the feature descriptor, in contrast to classical approaches (SIFT, etc.), where the detection phase precedes and differs from computing the descriptor. Our approach utilizes two CNN subnetworks, the first being a Siamese CNN and the second, consisting of dual non-weight-sharing CNNs. This allows simultaneous processing and fusion of the joint and disjoint cues in the multimodal image patches. The proposed approach is experimentally shown to outperform contemporary state-of-the-art schemes when applied to multiple datasets of multimodal images. It is also shown to provide repeatable feature points detections across multi-sensor images, outperforming state-of-the-art detectors. To the best of our knowledge, it is the first unified approach for the detection and matching of such images.