Discourse signals are often implicit, leaving it up to the interpreter to draw the required inferences. At the same time, discourse is embedded in a social context, meaning that interpreters apply their own assumptions and beliefs when resolving these inferences, leading to multiple, valid interpretations. However, current discourse data and frameworks ignore the social aspect, expecting only a single ground truth. We present the first discourse dataset with multiple and subjective interpretations of English conversation in the form of perceived conversation acts and intents. We carefully analyze our dataset and create computational models to (1) confirm our hypothesis that taking into account the bias of the interpreters leads to better predictions of the interpretations, (2) and show disagreements are nuanced and require a deeper understanding of the different contextual factors. We share our dataset and code at http://github.com/elisaF/subjective_discourse.