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Although the differential privacy (DP) of decentralized online learning has garnered considerable attention recently, existing algorithms are unsatisfactory due to their inability to achieve $(\epsilon, 0)$-DP over all $T$ rounds, recover the optimal regret in the non-private case, and maintain the lightweight computation under complex constraints. To address these issues, we first propose a new decentralized online learning algorithm satisfying $(\epsilon, 0)$-DP over $T$ rounds, and show that it can achieve $\widetilde{O}(n(\rho^{-1/4}+\epsilon^{-1}\rho^{1/4})\sqrt{T})$ and $\widetilde{O}(n (\rho^{-1/2}+\epsilon^{-1}))$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners and $\rho$ is the spectral gap of the communication matrix. As long as $\epsilon=\Omega(\sqrt{\rho})$, these bounds nearly match existing lower bounds in the non-private case, which implies that $(\epsilon, 0)$-DP of decentralized online learning may be ensured nearly for free. Our key idea is to design a block-decoupled accelerated gossip strategy that can be incorporated with the classical tree-based private aggregation, and also enjoys a faster average consensus among local learners. Furthermore, we develop a projection-free variant of our algorithm to keep the efficiency under complex constraints. As a trade-off, the above regret bounds degrade to $\widetilde{O}(n(T^{3/4}+\epsilon^{-1}T^{1/4}))$ and $\widetilde{O}(n(T^{2/3}+\epsilon^{-1}))$ respectively, which however are even better than the existing private centralized projection-free online algorithm.