Gossip protocols, also called rumor spreading or epidemic protocols, are widely used to disseminate information in massive peer-to-peer networks. These protocols are often claimed to guarantee privacy because of the uncertainty they introduce on the node that started the dissemination. But is that claim really true? Can one indeed start a gossip and safely hide in the crowd? This paper is the rst to study gossip protocols using a rigorous mathematical framework based on di erential privacy to determine the extent to which the source of a gossip can be traceable. Considering the case of a complete graph in which a subset of the nodes are curious, we derive matching lower and upper bounds on di erential privacy showing that some gossip protocols achieve strong privacy guarantees. Our results further reveal an interesting tension between privacy and dissemination speed: the standard "push" gossip protocol has very weak privacy guarantees, while the optimal guarantees are attained at the cost of a drastic increase in the spreading time. Yet, we show that it is possible to leverage the inherent randomness and partial observability of gossip protocols to achieve both fast dissemination speed and near-optimal privacy.