Obstructive sleep apnea (OSA) is a chronic and prevalent condition with well-established comorbidities. Due to limited diagnostic resources and high cost, a significant OSA population lives undiagnosed, and accurate and low-cost methods to screen for OSA are needed. We propose a novel screening method based on breathing sounds recorded with a smartphone and respiratory effort. Whole night recordings are divided into 30-s segments, each of which is classified for the presence or absence of OSA events by a multimodal deep neural network. Data fusion techniques were investigated and evaluated based on the apnea-hypopnea index estimated from whole night recordings. Real-world recordings made during home sleep apnea testing from 103 participants were used to develop and evaluate the proposed system. The late fusion system achieved the best sensitivity and specificity when screening for severe OSA, at 0.93 and 0.92, respectively. This offers the prospect of inexpensive OSA screening at home.