Respondent-driven sampling (RDS) is a popular method for sampling hard-to-survey populations that leverages social network connections through peer recruitment. While RDS is most frequently applied to estimate the prevalence of infections and risk behaviors of interest to public health, like HIV/AIDS or condom use, it is rarely used to draw inferences about the structural properties of social networks among such populations because it does not typically collect the necessary data. Drawing on recent advances in computer science, we introduce a set of data collection instruments and RDS estimators for network clustering, an important topological property that has been linked to a network's potential for diffusion of information, disease, and health behaviors. We use simulations to explore how these estimators, originally developed for random walk samples of computer networks, perform when applied to RDS samples with characteristics encountered in realistic field settings that depart from random walks. In particular, we explore the effects of multiple seeds, without vs. with replacement, branching chains, imperfect response rates, preferential recruitment, and misreporting of ties. We find that clustering coefficient estimators retain desirable properties in RDS samples. This paper takes an important step towards calculating network characteristics using non-traditional sampling methods, and it expands RDS's potential to tell researchers more about hidden populations and the social factors driving disease prevalence.