We report on a data-driven investigation aimed at understanding the dynamics of message spreading in a real-world dynamical network of human proximity. We use data collected by means of a proximity-sensing network of wearable sensors that we deployed at three different social gatherings, simultaneously involving several hundred individuals. We simulate a message spreading process over the recorded proximity network, focusing on both the topological and the temporal properties. We show that by using an appropriate technique to deal with the temporal heterogeneity of proximity events, a universal statistical pattern emerges for the delivery times of messages, robust across all the data sets. Our results are useful to set constraints for generic processes of data dissemination, as well as to validate established models of human mobility and proximity that are frequently used to simulate realistic behaviors.