Continuous cough monitors can greatly benefit doctors in home monitoring and treatment of respiratory diseases. Although many works propose algorithms to automate this task, they suffer of poor data privacy and short-term monitoring. Edge-AI is a promising paradigm to overcome these limitations by processing privacy-sensitive data close to their source. However, it presents challenges for the deployment of resource-demanding algorithms on constrained devices. In this work, we propose a hardwareaware methodology for developing a cough detection algorithm, analyzing design-time trade-offs for performance and energy. From audio and kinematic signals, our methodology aims at optimal features via Recursive Feature Elimination with Cross-Validation (RFECV), exploiting the explainability of the selected XGB model. Additionally, it analyzes the use of Mel spectrogram features, instead of the common MFCC. Moreover, a set of hyperparameters for a multimodal implementation of the classifier is explored. Finally, it evaluates the performance based on clinically relevant event-based metrics. The methodology proposes a novel structured approach to efficiently deploy AI on the edge, preserving data privacy. We apply our methodology to develop Cough-E, an energy-efficient, multimodal, and edge AI cough detector. It exploits audio and kinematic data in two distinct models, cooperating for a balanced energy and performance trade-off. We demonstrate that our algorithm can be executed in real-time on an ARM Cortex M33 microcontroller. Cough-E achieves a 70.56% energy saving compared to the audio-only approach, for a 1.26% relative performance drop, resulting in a 0.78 F1-score. Both Cough-E and the edge-aware model optimization methodology are available as open-source code. This approach demonstrates the benefits of the proposed hardware-aware methodology to enable privacy-preserving cough monitors on the edge, paving the way to efficient cough monitoring.