In this work, we carry out the first, in-depth, privacy analysis of Decentralized Learning -- a collaborative machine learning framework aimed at circumventing the main limitations of federated learning. We identify the decentralized learning properties that affect users' privacy and we introduce a suite of novel attacks for both passive and active decentralized adversaries. We demonstrate that, contrary to what is claimed by decentralized learning proposers, decentralized learning does not offer any security advantages over more practical approaches such as federated learning. Rather, it tends to degrade users' privacy by increasing the attack surface and enabling any user in the system to perform powerful privacy attacks such as gradient inversion, and even gain full control over honest users' local model. We also reveal that, given the state of the art in protections, privacy-preserving configurations of decentralized learning require abandoning any possible advantage over the federated setup, completely defeating the objective of the decentralized approach. 17 pages