In this paper, we present AltVoice -- a system designed to help user's protect their privacy when using remotely accessed voice services. The system allows a user to conceal their true voice identity information with no cooperation from the remote voice service: AltVoice re-synthesizes user's spoken audio to sound as if it has been spoken by a different, private identity. The system converts audio to its textual representation at its midpoint, and thus removes any linkage between the user's voice and the generated private voices. We implement AltVoice and we propose six different methods to generate private voice identities, each is based on a user-known secret. We identify the system's trade-offs, and we investigate them for each of the proposed identity generation methods. Specifically, we investigate generated voices' diversity, word error rate, perceived speech quality and the success of attackers under privacy compromise and authentication compromise attack scenarios. Our results show that AltVoice-generated voices are not easily linked to original users, enabling users to protect themselves from voice data leakages and allowing for the revocability of (generated) voice data; akin to using passwords. However the results also show further work is needed on ensuring that the produced audio is natural, and that identities of private voices are distinct from one another. We discuss the future steps into improving AltVoice and the new implications that its existence has for the creations of remotely accessed voice services.