Smart devices serviced by large-scale AI models necessitates user data transfer to the cloud for inference. For speech applications, this means transferring private user information, e.g., speaker identity. Our paper proposes a privacy-enhancing framework that targets speaker identity anonymization while preserving speech recognition accuracy for our downstream task~-~Automatic Speech Recognition (ASR). The proposed framework attaches flexible gradient reversal based speaker adversarial layers to target layers within an ASR model, where speaker adversarial training anonymizes acoustic embeddings generated by the targeted layers to remove speaker identity. We propose on-device deployment by execution of initial layers of the ASR model, and transmitting anonymized embeddings to the cloud, where the rest of the model is executed while preserving privacy. Experimental results show that our method efficiently reduces speaker recognition relative accuracy by 33%, and improves ASR performance by achieving 6.2% relative Word Error Rate (WER) reduction.