In recommender systems, users' behavior data are driven by the interactions of user-item latent factors. To improve recommendation effectiveness and robustness, recent advances focus on latent factor disentanglement via variational inference. Despite significant progress, uncovering the underlying interactions, i.e., dependencies of latent factors, remains largely neglected by the literature. To bridge the gap, we investigate the joint disentanglement of user-item latent factors and the dependencies between them, namely latent structure learning. We propose to analyze the problem from the causal perspective, where a latent structure should ideally reproduce observational interaction data, and satisfy the structure acyclicity and dependency constraints, i.e., causal prerequisites. We further identify the recommendation-specific challenges for latent structure learning, i.e., the subjective nature of users' minds and the inaccessibility of private/sensitive user factors causing universally learned latent structure to be suboptimal for individuals. To address these challenges, we propose the personalized latent structure learning framework for recommendation, namely PlanRec, which incorporates 1) differentiable Reconstruction, Dependency, and Acyclicity regularizations to satisfy the causal prerequisites; 2) Personalized Structure Learning (PSL) which personalizes the universally learned dependencies through probabilistic modeling; and 3) uncertainty estimation which explicitly measures the uncertainty of structure personalization, and adaptively balances personalization and shared knowledge for different users. We conduct extensive experiments on two public benchmark datasets from MovieLens and Amazon, and a large-scale industrial dataset from Alipay. Empirical studies validate that PlanRec discovers effective shared/personalized structures, and successfully balances shared knowledge and personalization via rational uncertainty estimation.