Respondent-driven sampling (RDS) is a network sampling method commonly used to access hidden populations when conventional sampling techniques are not possible. Data from RDS surveys inform key policy and resource allocation decisions, and in particular population size estimations are essential to understand counts of at-risk individuals. Successive sampling population size estimation (SS-PSE) is a commonly used method to estimate population size from RDS surveys, in which the decrease in social network size of participants over the study period is used to gauge the sample fraction. However, SS-PSE relies on self-reported social network sizes, which are subject to missingness, misreporting, and bias. We present a modification to the SS-PSE methodology that jointly models the effective social network size of each individual along with population size in a Bayesian framework. The model for effective network size, which we call visibility to reflect its usage as a proxy for inclusion probability, incorporates a measurement error model for self-reported network size. We present the imputed visibility SS-PSE framework and demonstrate its utility on three RDS datasets from Kosovo.