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Breathing biomarkers, such as breathing rate, fractional inspiratory time, and inhalation-exhalation ratio, are vital for monitoring the user's health and well-being. Accurate estimation of such biomarkers requires breathing phase detection, i.e., inhalation and exhalation. However, traditional breathing phase monitoring relies on uncomfortable equipment, e.g., chestbands. Smartphone acoustic sensors have shown promising results for passive breathing monitoring during sleep or guided breathing. However, detecting breathing phases using acoustic data can be challenging for various reasons. One of the major obstacles is the complexity of annotating breathing sounds due to inaudible parts in regular breathing and background noises. This paper assesses the potential of using smartphone acoustic sensors for passive unguided breathing phase monitoring in a natural environment. We address the annotation challenges by developing a novel variant of the teacher-student training method for transferring knowledge from an inertial sensor to an acoustic sensor, eliminating the need for manual breathing sound annotation by fusing signal processing with deep learning techniques. We train and evaluate our model on the breathing data collected from 131 subjects, including healthy individuals and respiratory patients. Experimental results show that our model can detect breathing phases with 77.33% accuracy using acoustic sensors. We further present an example use-case of breathing phase-detection by first estimating the biomarkers from the estimated breathing phases and then using these biomarkers for pulmonary patient detection. Using the detected breathing phases, we can estimate fractional inspiratory time with 92.08% accuracy, the inhalation-exhalation ratio with 86.76% accuracy, and the breathing rate with 91.74% accuracy. Moreover, we can distinguish respiratory patients from healthy individuals with up to 76% accuracy. This paper is the first to show the feasibility of detecting regular breathing phases towards passively monitoring respiratory health and well-being using acoustic data captured by a smartphone.