We study multitask online learning in a setting where agents can only exchange information with their neighbors on an arbitrary communication network. We introduce MT-CO\textsubscript{2}OL, a decentralized algorithm for this setting whose regret depends on the interplay between the task similarities and the network structure. Our analysis shows that the regret of MT-CO\textsubscript{2}OL is never worse (up to constants) than the bound obtained when agents do not share information. On the other hand, our bounds significantly improve when neighboring agents operate on similar tasks. In addition, we prove that our algorithm can be made differentially private with a negligible impact on the regret. Finally, we provide experimental support for our theory.