The intrinsic hardware imperfection of WiFi chipsets manifests itself in the transmitted signal, leading to a unique radiometric fingerprint. This fingerprint can be used as an additional means of authentication to enhance security. In fact, recent works propose practical fingerprinting solutions that can be readily implemented in commercial-off-the-shelf devices. In this paper, we prove analytically and experimentally that these solutions are highly vulnerable to impersonation attacks. We also demonstrate that such a unique device-based signature can be abused to violate privacy by tracking the user device, and, as of today, users do not have any means to prevent such privacy attacks other than turning off the device. We propose RF-Veil, a radiometric fingerprinting solution that not only is robust against impersonation attacks but also protects user privacy by obfuscating the radiometric fingerprint of the transmitter for non-legitimate receivers. Specifically, we introduce a randomized pattern of phase errors to the transmitted signal such that only the intended receiver can extract the original fingerprint of the transmitter. In a series of experiments and analyses, we expose the vulnerability of adopting naive randomization to statistical attacks and introduce countermeasures. Finally, we show the efficacy of RF-Veil experimentally in protecting user privacy and enhancing security. More importantly, our proposed solution allows communicating with other devices, which do not employ RF-Veil.