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In response to the challenges presented by high reviewer workloads in traditional panel reviews and increasing numbers of submitted proposals, ALMA implemented distributed peer review to assess the majority of proposals submitted to the Cycle 8 Main Call. In this paper, we present an analysis of this review process. Over 1000 reviewers participated in the process to review 1497 proposals, making it the largest implementation of distributed peer review to date in astronomy, and marking the first time this process has been used to award the majority of observing time at an observatory. We describe the process to assign proposals to reviewers, analyze the nearly 15,000 ranks and comments submitted by reviewers to identify any trends and systematics, and gather feedback on the process from reviewers and Principal Investigators (PIs) through surveys. Approximately 90% of the proposal assignments were aligned with the expertise of the reviewer, as measured both by the expertise keywords provided by the reviewers and the reviewers' self-assessment of their expertise on their assigned proposals. PIs rated 73% of the individual review comments as helpful, and even though the reviewers had a broad range of experience levels, PIs rated the quality of the comments received from students and senior researchers similarly. The primary concerns raised by PIs were the quality of some reviewer comments and high dispersions in the ranks. The ranks and comments are correlated with various demographics to identify the main areas in which the review process can be improved in future cycles.