Open-domain dialogue systems aim to converse with humans through text, and dialogue research has heavily relied on benchmark datasets. In this work, we observe the overlapping problem in DailyDialog and OpenSubtitles, two popular open-domain dialogue benchmark datasets. Our systematic analysis then shows that such overlapping can be exploited to obtain fake state-of-the-art performance. Finally, we address this issue by cleaning these datasets and setting up a proper data processing procedure for future research.