Long-term quantification of asthmatic wheezing envisions an m-Health sensor system consisting of a smartphone and a body-worn wireless acoustic sensor. As both devices are power constrained, the main criterion guiding the system design comes down to minimization of power consumption, while retaining sufficient respiratory sound classification accuracy (i.e., wheeze detection). Crucial for assessment of the system-level power consumption is the understanding of trade-off between power cost of computationally intensive local processing and communication. Therefore, we analyze power requirements of signal acquisition, processing, and communication in three typical operating scenarios: (1) streaming of uncompressed respiratory signal to a smartphone for classification, (2) signal streaming utilizing compressive sensing (CS) for reduction of data rate, and (3) respiratory sound classification onboard the wearable sensor. Study shows that the third scenario featuring the lowest communication cost enables the lowest total sensor system power consumption ranging from 328 to 428 μW. In such scenario, 32-bit ARM Cortex M3/M4 cores typically embedded within Bluetooth 4 SoC modules feature the optimal trade-off between onboard classification performance and consumption. On the other hand, study confirms that CS enables the most power-efficient design of the wearable sensor (216 to 357 μW) in the compressed signal streaming, the second scenario. In such case, a single low-power ARM Cortex-A53 core is sufficient for simultaneous real-time CS reconstruction and classification on the smartphone, while keeping the total system power within budget for uncompressed streaming.