This paper presents a dual-channel speech enhancement framework that effectively integrates deep neural network (DNN) mask estimators. Our framework follows a beamforming-plus-postfiltering approach intended for noise reduction on dual-microphone smartphones. An extended Kalman filter is used for the estimation of the relative acoustic channel between microphones, while the noise estimation is performed using a speech presence probability estimator. We propose the use of a DNN estimator to improve the prediction of the speech presence probabilities without making any assumption about the statistics of the signals. We evaluate and compare different dual-channel features to improve the accuracy of this estimator, including the power and phase difference between the speech signals at the two microphones. The proposed integrated scheme is evaluated in different reverberant and noisy environments when the smartphone is used in both close- and far-talk positions. The experimental results show that our approach achieves significant improvements in terms of speech quality, intelligibility, and distortion when compared to other approaches based only on statistical signal processing.