Cytosine methylation is an epigenetic mechanism in eukaryotes that is often associated with stable transcriptional silencing, such as in X-chromosome inactivation and genomic imprinting. Aberrant methylation patterns occur in several inherited human diseases and in many cancers. To understand how methylated and unmethylated states of cytosine residues are transmitted during DNA replication, we develop a population-epigenetic model of DNA methylation dynamics. The model is informed by our observation that de novo methylation can occur on the daughter strand while leaving the opposing cytosine unmethylated, as revealed by the patterns of methylation on the two complementary strands of individual DNA molecules. Under our model, we can infer site-specific rates of both maintenance and de novo methylation, values that determine the fidelity of methylation inheritance, from double-stranded methylation data. This approach can be used for populations of cells obtained from individuals without the need for cell culture. We use our method to infer cytosine methylation rates at several sites within the promoter of the human gene FMR1.