Distributed online learning is gaining increased traction due to its unique ability to process large-scale datasets and streaming data. To address the growing public awareness and concern on privacy protection, plenty of algorithms have been proposed to enable differential privacy in distributed online optimization and learning. However, these algorithms often face the dilemma of trading learning accuracy for privacy. By exploiting the unique characteristics of online learning, this paper proposes an approach that tackles the dilemma and ensures both differential privacy and learning accuracy in distributed online learning. More specifically, while ensuring a diminishing expected instantaneous regret, the approach can simultaneously ensure a finite cumulative privacy budget, even in the infinite time horizon. To cater for the fully distributed setting, we adopt the local differential-privacy framework, which avoids the reliance on a trusted data curator that is required in the classic "centralized" (global) differential-privacy framework. To the best of our knowledge, this is the first algorithm that successfully ensures both rigorous local differential privacy and learning accuracy. The effectiveness of the proposed algorithm is evaluated using machine learning tasks, including logistic regression on the the "mushrooms" datasets and CNN-based image classification on the "MNIST" and "CIFAR-10" datasets.