In surveillance studies of periodontal disease, the relationship between disease and other health and socioeconomic conditions is of key interest. To determine whether or not a patient has periodontal disease, multiple clinical measurements are taken at the tooth-level. Moreover, patients have varying number of teeth, with those that are more prone to the disease having fewer teeth compared to those with good oral health. Such dependence between the outcome of interest and cluster size (number of teeth) is called informative cluster size and results obtained from fitting conventional marginal models can be biased. We propose a novel method to jointly analyze multiple correlated binary outcomes for clustered data with informative cluster size using the class of generalized estimating equations (GEE) with cluster-specific weights. We compare our proposed multivariate outcome cluster-weighted GEE results to those from the convectional GEE using the baseline data from Veterans Affairs Dental Longitudinal Study. In an extensive simulation study, we show that our proposed method yields estimates with minimal relative biases and excellent coverage probabilities.