Distributed online learning has been proven extremely effective in solving large-scale machine learning problems over streaming data. However, information sharing between learners in distributed learning also raises concerns about the potential leakage of individual learners' sensitive data. To mitigate this risk, differential privacy, which is widely regarded as the "gold standard" for privacy protection, has been widely employed in many existing results on distributed online learning. However, these results often face a fundamental tradeoff between learning accuracy and privacy. In this paper, we propose a locally differentially private gradient tracking based distributed online learning algorithm that successfully circumvents this tradeoff. We prove that the proposed algorithm converges in mean square to the exact optimal solution while ensuring rigorous local differential privacy, with the cumulative privacy budget guaranteed to be finite even when the number of iterations tends to infinity. The algorithm is applicable even when the communication graph among learners is directed. To the best of our knowledge, this is the first result that simultaneously ensures learning accuracy and rigorous local differential privacy in distributed online learning over directed graphs. We evaluate our algorithm's performance by using multiple benchmark machine-learning applications, including logistic regression of the "Mushrooms" dataset and CNN-based image classification of the "MNIST" and "CIFAR-10" datasets, respectively. The experimental results confirm that the proposed algorithm outperforms existing counterparts in both training and testing accuracies.