In medical studies, paired binary responses are often observed for each study subject over timepoints or clusters. A primary interest is to investigate how the bivariate association and marginal univariate risks are affected by repeated measurements on each subject. To achieve this we propose a very general class of semiparametric bivariate binary models. The subject-specific effects involved in the bivariate log odds ratio and the univariate logit components are assumed to follow a nonparametric Dirichlet process (DP). We propose a hybrid method to draw model-based inferences. In the framework of the proposed hybrid method, estimation of parameters is done by implementing the Monte Carlo expectation-maximization algorithm. The proposed methodology is illustrated through a study on the effectiveness of tibolone for reducing menopausal problems experienced by Indian women. A simulation study is also conducted to evaluate the efficiency of the new methodology. © 2006, The International Biometric Society.