When people communicate in their native languages using machine translation, they face various problems in constructing common ground. This study, based on the Language Grid framework, investigates the difficulties of constructing common ground when pairs and triads communicate using machine translation. We compare referential communication of pairs and triads under two conditions: in their shared second language (English) and in their native languages using machine translation. Consequently, to support natural referring behaviour in machine translation mediated communication between pairs, our study suggests the importance of resolving the asymmetries and inconsistencies caused by machine translations. Furthermore, to successfully build common ground among triads, it is important for addressees to be able to monitor what is going on between a speaker and other addressees. The findings serve as a basis for designing future machine translation embedded communication systems. The proposed design implications, in particular, are fed back to the Language Grid development process and incorporated into the recent Language Grid Toolbox.