Many classes of network growth models have been proposed in the literature for capturing real-world complex networks. Existing research primarily focuses on global characteristics of these models, e.g., degree distribution. We aim to shift the focus towards studying the network growth dynamics from the perspective of individual nodes. In this paper, we study how a metric for node influence in network growth models behaves over time as the network evolves. This metric, which we call node visibility, captures the probability of the node to form new connections. First, we conduct an investigation on three popular network growth models -- preferential attachment, additive, and multiplicative fitness models; and primarily look into the "influential nodes" or "leaders" to understand how their visibility evolves over time. Subsequently, we consider a generic fitness model and observe that the multiplicative model strikes a balance between allowing influential nodes to maintain their visibility, while at the same time making it possible for new nodes to gain visibility in the network. Finally, we observe that a spatial growth model with multiplicative fitness can curtail the global reach of influential nodes, thereby allowing the emergence of a multiplicity of "local leaders" in the network.