Animal societies rely on interactions between group members to effectively communicate and coordinate their actions. To date, the transmission properties of interaction networks formed by direct physical contacts have been extensively studied for many animal societies and in all cases found to inhibit spreading. Such direct interactions do not, however, represent the only viable pathways. When spreading agents can persist in the environment, indirect transmission via 'same-place, different-time' spatial coincidences becomes possible. Previous studies have neglected these indirect pathways and their role in transmission. Here, we use rock ant colonies, a model social species whose flat nest geometry, coupled with individually tagged workers, allowed us to build temporally and spatially explicit interaction networks in which edges represent either direct physical contacts or indirect spatial coincidences. We show how the addition of indirect pathways allows the network to enhance or inhibit the spreading of different types of agent. This dual-functionality arises from an interplay between the interaction-strength distribution generated by the ants' movement and environmental decay characteristics of the spreading agent. These findings offer a general mechanism for understanding how interaction patterns might be tuned in animal societies to control the simultaneous transmission of harmful and beneficial agents.