Online social networks play an increasingly important role in communication between friends, colleagues, business partners, and family members. This development sparked public and scholarly debate about how these new platforms affect dynamics of cultural diversity. Formal models of cultural dissemination are powerful tools to study dynamics of cultural diversity but they are based on assumptions that represent traditional dyadic, face-To-face communication, rather than communication in online social networks. Unlike in models of face-To-face communication, where actors update their cultural traits after being influenced by one of their network contacts, communication in online social networks is often characterized by a one-To-many structure, in that users emit messages directly to a large number of network contacts. Using analytical tools and agent-based simulation, we show that this seemingly subtle difference can have profound implications for emergent dynamics of cultural dissemination. In particular, we show that within the framework of our model online communication fosters cultural diversity to a larger degree than offline communication and it increases chances that individuals and subgroups become culturally isolated from their network contacts.