Electrodermal activity (EDA) is a physiological signal that can be used to infer humans' affective states and stress levels. EDA can nowadays be monitored using unobtrusive wearable devices, such as smartwatches, and leveraged in personal informatics systems. A still largely uncharted issue concerning EDA is the impact on real applications of potential differences observable on signals measured concurrently on the left and right side of the human body. This phenomenon, called lateralization, originates from the distinct functions that the brain's left and right hemispheres exert on EDA. In this work, we address this issue by examining the impact of EDA lateralization in two classification tasks: a cognitive load recognition task executed in the lab and a sleep monitoring task in a real-world setting. We implement a machine learning pipeline to compare the performance obtained on both classification tasks using EDA data collected from the left and right sides of the body. Our results show that using EDA from the side that is not associated with the specific hemisphere activation leads to a significant decline in performance for the considered classification tasks. This finding highlights that researchers and practitioners relying on EDA data should consider possible EDA lateralization effects when deciding on sensor placement.