Federated recommendation is a new Internet service architecture that aims to provide privacy-preserving recommendation services in federated settings. Existing solutions are used to combine distributed recommendation algorithms and privacy-preserving mechanisms. Thus it inherently takes the form of heavyweight models at the server and hinders the deployment of on-device intelligent models to end-users. This paper proposes a novel Personalized Federated Recommendation (PFedRec) framework to learn many user-specific lightweight models to be deployed on smart devices rather than a heavyweight model on a server. Moreover, we propose a new dual personalization mechanism to effectively learn fine-grained personalization on both users and items. The overall learning process is formulated into a unified federated optimization framework. Specifically, unlike previous methods that share exactly the same item embeddings across users in a federated system, dual personalization allows mild finetuning of item embeddings for each user to generate user-specific views for item representations which can be integrated into existing federated recommendation methods to gain improvements immediately. Experiments on multiple benchmark datasets have demonstrated the effectiveness of PFedRec and the dual personalization mechanism. Moreover, we provide visualizations and in-depth analysis of the personalization techniques in item embedding, which shed novel insights on the design of recommender systems in federated settings. The code is available.