Voice User Interfaces (VUIs) are increasingly popular and built into smartphones, home assistants, and Internet of Things (IoT) devices. Despite offering an always-on convenient user experience, VUIs raise new security and privacy concerns for their users. In this paper, we focus on attribute inference attacks in the speech domain, demonstrating the potential for an attacker to accurately infer a target user's sensitive and private attributes (e.g. their emotion, sex, or health status) from deep acoustic models. To defend against this class of attacks, we design, implement, and evaluate a user-configurable, privacy-aware framework for optimizing speech-related data sharing mechanisms. Our objective is to enable primary tasks such as speech recognition and user identification, while removing sensitive attributes in the raw speech data before sharing it with a cloud service provider. We leverage disentangled representation learning to explicitly learn independent factors in the raw data. Based on a user's preferences, a supervision signal informs the filtering out of invariant factors while retaining the factors reflected in the selected preference. Our experimental evaluation over five datasets shows that the proposed framework can effectively defend against attribute inference attacks by reducing their success rates to approximately that of guessing at random, while maintaining accuracy in excess of 99% for the tasks of interest. We conclude that negotiable privacy settings enabled by disentangled representations can bring new opportunities for privacy-preserving applications.