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Motivated by information sharing in online platforms, we study repeated persuasion between a sender and a stream of receivers where at each time, the sender observes a payoff-relevant state drawn independently and identically from an unknown distribution, and shares state information with the receivers who each choose an action. The sender seeks to persuade the receivers into taking actions aligned with the sender's preference by selectively sharing state information. However, in contrast to the standard models, neither the sender nor the receivers know the distribution, and the sender has to persuade while learning the distribution on the fly. We study the sender's learning problem of making persuasive action recommendations to achieve low regret against the optimal persuasion mechanism with the knowledge of the distribution. To do this, we first propose and motivate a persuasiveness criterion for the unknown distribution setting that centers robustness as a requirement in the face of uncertainty. Our main result is an algorithm that, with high probability, is robustly-persuasive and achieves $O(\sqrt{T\log T})$ regret, where $T$ is the horizon length. Intuitively, at each time our algorithm maintains a set of candidate distributions, and chooses a signaling mechanism that is simultaneously persuasive for all of them. Core to our proof is a tight analysis about the cost of robust persuasion, which may be of independent interest. We further prove that this regret order is optimal (up to logarithmic terms) by showing that no algorithm can achieve regret better than $\Omega(\sqrt{T})$., Comment: Accepted at Operations Research. Preliminary version appeared as an extended abstract in EC 2021