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Normalization is an important but understudied challenge in privacy-related application domains such as federated learning (FL), differential privacy (DP), and differentially private federated learning (DP-FL). While the unsuitability of batch normalization for these domains has already been shown, the impact of other normalization methods on the performance of federated or differentially private models is not well-known. To address this, we draw a performance comparison among layer normalization (LayerNorm), group normalization (GroupNorm), and the recently proposed kernel normalization (KernelNorm) in FL, DP, and DP-FL settings. Our results indicate LayerNorm and GroupNorm provide no performance gain compared to the baseline (i.e. no normalization) for shallow models in FL and DP. They, on the other hand, considerably enhance the performance of shallow models in DP-FL and deeper models in FL and DP. KernelNorm, moreover, significantly outperforms its competitors in terms of accuracy and convergence rate (or communication efficiency) for both shallow and deeper models in all considered learning environments. Given these key observations, we propose a kernel normalized ResNet architecture called KNResNet-13 for differentially private learning. Using the proposed architecture, we provide new state-of-the-art accuracy values on the CIFAR-10 and Imagenette datasets, when trained from scratch.