We have studied the emergence of mutual synchronization and activity propagation in coupled neural networks from rat cortical cells grown on a micro-electrode array for parallel activity recording of dozens of neurons. The activity of each sub-network by itself is marked by the formation of synchronized bursting events (SBE) - short time windows of rapid neuronal firing. The joint activity of two coupled networks is characterized by the formation of mutual synchronization, i.e. the formation of SBE whose activity starts at one sub-network and then propagates to the other. The sub-networks switch roles in initiating the mutual SBE. However, spontaneous propagation (initiation) asymmetry emerges - one of the sub-networks takes on the role of initiating substantially more mutual SBE than the other, despite the fact that the two are engineered to be similar in size and cell density. Analysis of the interneuron correlations in the SBE also reveals the emergence of activity (function) asymmetry - one sub-network develops a more organized structure of correlations. We also show activity propagation and mutual synchronization in four coupled networks. Using computer simulations, we propose that the function asymmetry reflects asymmetry between the internal connectivity of the two networks, whereas the propagation asymmetry reflects asymmetry in the connectivity between the sub-networks. These results agree with the experimental findings that the initiation and function asymmetry can be separately regulated, which implies that information transfer (activity propagation) and information processing (function) can be regulated separately in coupled neural networks.