In this paper, we describe research exploring the effect of behavioral and visual realism of avatars on users' social copresence in emotionally engaged conversations conducted via a simulated mobile video telephone. We offer an elaborated definition of Social Copresence to better measure users' engagement with conversational partners in social interactions that do not involve specific tasks or concrete outcomes. We investigate ways to secure mobile telephone users' anonymity while preserving their most important nonverbal affective behaviors. Experimental results with 180 participants using different combinations of static and dynamic, high and low iconic (both video and graphically animated) avatars show increased Social Copresence with dynamic high-iconic (similar to the human communicator) avatars incorporating correct facial expressions, even when these are presented on the small screen of mobile telephones in such a way that individual identities are masked. The results point to an economical combination of behavioral and iconic realism of avatars that produces maximum emotional engagement in anonymous social interactions using mobile video telephones.