The Monte Carlo method is an established tool that is often used to evaluate the uncertainty of measurements. For computationally challenging problems, Monte Carlo uncertainty-analyses are typically distributed across multiple processes on a multi-node cluster or supercomputer. Additionally, results from previous uncertainty analyses are often used in further analyses, leading to uncertainty analyses that are temporally distributed as well. To accurately capture the uncertainty of the output quantity of interest, Monte Carlo sample distributions must be treated consistently, using reproducible replicates, throughout the entire analysis. We highlight the need and importance of consistent and accurate Monte Carlo methods in distributed uncertainty analyses, recommend an implementation to achieve the needed consistency in distributed analyses, and discuss methods to evaluate the accuracy of implementations.