In recent years, Large Language Models (LLMs) have demonstrated remarkable abilities in various natural language processing tasks. However, adapting these models to specialized domains using private datasets stored on resource-constrained edge devices, such as smartphones and personal computers, remains challenging due to significant privacy concerns and limited computational resources. Existing model adaptation methods either compromise data privacy by requiring data transmission or jeopardize model privacy by exposing proprietary LLM parameters. To address these challenges, we propose Prada, a novel privacy-preserving and efficient black-box LLM adaptation system using private on-device datasets. Prada employs a lightweight proxy model fine-tuned with Low-Rank Adaptation (LoRA) locally on user devices. During inference, Prada leverages the logits offset, i.e., difference in outputs between the base and adapted proxy models, to iteratively refine outputs from a remote black-box LLM. This offset-based adaptation approach preserves both data privacy and model privacy, as there is no need to share sensitive data or proprietary model parameters. Furthermore, we incorporate speculative decoding to further speed up the inference process of Prada, making the system practically deployable on bandwidth-constrained edge devices, enabling a more practical deployment of Prada. Extensive experiments on various downstream tasks demonstrate that Prada achieves performance comparable to centralized fine-tuning methods while significantly reducing computational overhead by up to 60% and communication costs by up to 80%.