The dramatically decreasing costs of DNA sequencing have triggered more than a million humans to have their genotypes sequenced. Moreover, these individuals increasingly make their genomic data publicly available, thereby creating privacy threats for themselves and their relatives because of their DNA similarities. More generally, an entity that gains access to a significant fraction of sequenced genotypes might be able to infer even the genomes of unsequenced individuals. In this paper, we propose a simulation-based model for quantifying the impact of continuously sequencing and publicizing personal genomic data on a population's genomic privacy. Our simulation probabilistically models data sharing and takes into account events such as migration and interracial mating. We exemplarily instantiate our simulation with a sample population of 1,000 individuals and evaluate the privacy under multiple settings over 6,000 genomic variants and a subset of phenotype-related variants. Our findings demonstrate that an increasing sharing rate in the future entails a substantial negative effect on the privacy of all older generations. Moreover, we find that mixed populations face a less severe erosion of privacy over time than more homogeneous populations. Finally, we demonstrate that genomic-data sharing can be much more detrimental for the privacy of the phenotype-related variants.