The increasing popularity of autonomous and remotely-piloted drones have paved the way for several use-cases, e.g., merchandise delivery and surveillance. In many scenarios, estimating with zero-touch the weight of the payload carried by a drone before its physical approach could be attractive, e.g., to provide an early tampering detection. In this paper, we investigate the possibility to remotely detect the weight of the payload carried by a commercial drone by analyzing its acoustic fingerprint. We characterize the difference in the thrust needed by the drone to carry different payloads, resulting in significant variations of the related acoustic fingerprint. We applied the above findings to different use-cases, characterized by different computational capabilities of the detection system. Results are striking: using the Mel-Frequency Cepstral Coefficients (MFCC) components of the audio signal and different Support Vector Machine (SVM) classifiers, we achieved a minimum classification accuracy of 98% in the detection of the specific payload class carried by the drone, using an acquisition time of 0.25 s---performances improve when using longer time acquisitions. All the data used for our analysis have been released as open-source, to enable the community to validate our findings and use such data as a ready-to-use basis for further investigations.