We study the interplay between local differential privacy (LDP) and robustness to Huber corruption and possibly heavy-tailed rewards in the context of multi-armed bandits (MABs). We consider two different practical settings: LDP-then-Corruption (LTC) where each user's locally private response might be further corrupted during the data collection process, and Corruption-then-LDP (CTL) where each user's raw data may be corrupted such that the LDP mechanism will only be applied to the corrupted data. To start with, we present the first tight characterization of the mean estimation error in high probability under both LTC and CTL settings. Leveraging this new result, we then present an almost tight characterization (up to log factor) of the minimax regret in online MABs and sub-optimality in offline MABs under both LTC and CTL settings, respectively. Our theoretical results in both settings are also corroborated by a set of systematic simulations. One key message in this paper is that LTC is a more difficult setting that leads to a worse performance guarantee compared to the CTL setting (in the minimax sense). Our sharp understanding of LTC and CTL also naturally allows us to give the first tight performance bounds for the most practical setting where corruption could happen both before and after the LDP mechanism. As an important by-product, we also give the first correct and tight regret bound for locally private and heavy-tailed online MABs, i.e., without Huber corruption, by identifying a fundamental flaw in the state-of-the-art.