With the development of mobile communication technology and the proliferation of the number of Internet of Things (IoT) terminal devices, a large amount of data and intelligent applications are emerging at the edge of the Internet, giving rise to the demand for edge intelligence. In this context, federated learning, as a new distributed machine learning method, becomes one of the key technologies to realize edge intelligence. Traditional edge intelligence networks usually rely on terrestrial communication base stations as parameter servers to manage communication and computation tasks among devices. However, this fixed infrastructure is difficult to adapt to the complex and ever-changing heterogeneous network environment. With its high degree of flexibility and mobility, the introduction of unmanned aerial vehicles (UAVs) into the federated learning framework can provide enhanced communication, computation, and caching services in edge intelligence networks, but the limited communication bandwidth and unreliable communication environment increase system uncertainty and may lead to a decrease in overall energy efficiency. To address the above problems, this paper designs a UAV-assisted federated learning with a privacy-preserving and efficient data sharing method, Communication-efficient and Privacy-protection for FL (CP-FL). A network-sparsifying pruning training method based on a channel importance mechanism is proposed to transform the pruning training process into a constrained optimization problem. A quantization-aware training method is proposed to automate the learning of quantization bitwidths to improve the adaptability between features and data representation accuracy. In addition, differential privacy is applied to the uplink data on this basis to further protect data privacy. After the model parameters are aggregated on the pilot UAV, the model is subjected to knowledge distillation to reduce the amount of downlink data without affecting the utility. Experiments on real-world datasets validate the effectiveness of the scheme. The experimental results show that compared with other federated learning frameworks, the CP-FL approach can effectively mitigate the communication overhead, as well as the computation overhead, and has the same outstanding advantage in terms of the balance between privacy and usability in differential privacy preservation.