Graph Neural Networks (GNNs) have demonstrated superior performance in learning node representations for various graph inference tasks. However, learning over graph data can raise privacy concerns when nodes represent people or human-related variables that involve sensitive or personal information. While numerous techniques have been proposed for privacy-preserving deep learning over non-relational data, there is less work addressing the privacy issues pertained to applying deep learning algorithms on graphs. In this paper, we study the problem of node data privacy, where graph nodes have potentially sensitive data that is kept private, but they could be beneficial for a central server for training a GNN over the graph. To address this problem, we develop a privacy-preserving, architecture-agnostic GNN learning algorithm with formal privacy guarantees based on Local Differential Privacy (LDP). Specifically, we propose an LDP encoder and an unbiased rectifier, by which the server can communicate with the graph nodes to privately collect their data and approximate the GNN's first layer. To further reduce the effect of the injected noise, we propose to prepend a simple graph convolution layer, called KProp, which is based on the multi-hop aggregation of the nodes' features acting as a denoising mechanism. Finally, we propose a robust training framework, in which we benefit from KProp's denoising capability to increase the accuracy of inference in the presence of noisy labels. Extensive experiments conducted over real-world datasets demonstrate that our method can maintain a satisfying level of accuracy with low privacy loss.