While Graph Neural Networks (GNNs) are remarkably successful in a variety of high-impact applications, we demonstrate that, in link prediction, the common practices of including the edges being predicted in the graph at training and/or test have outsized impact on the performance of low-degree nodes. We theoretically and empirically investigate how these practices impact node-level performance across different degrees. Specifically, we explore three issues that arise: (I1) overfitting; (I2) distribution shift; and (I3) implicit test leakage. The former two issues lead to poor generalizability to the test data, while the latter leads to overestimation of the model's performance and directly impacts the deployment of GNNs. To address these issues in a systematic way, we introduce an effective and efficient GNN training framework, SpotTarget, which leverages our insight on low-degree nodes: (1) at training time, it excludes a (training) edge to be predicted if it is incident to at least one low-degree node; and (2) at test time, it excludes all test edges to be predicted (thus, mimicking real scenarios of using GNNs, where the test data is not included in the graph). SpotTarget helps researchers and practitioners adhere to best practices for learning from graph data, which are frequently overlooked even by the most widely-used frameworks. Our experiments on various real-world datasets show that SpotTarget makes GNNs up to 15x more accurate in sparse graphs, and significantly improves their performance for low-degree nodes in dense graphs.