Discourse on social media platforms is often plagued by acute polarization, with different camps promoting different perspectives on the issue at hand—compare, for example, the differences in the liberal and conservative discourse on the U.S. immigration debate. A large body of research has studied this phenomenon by focusing on the affiliation of groups and individuals. We propose a new finer-grained perspective: studying the impartiality of individual messages. While the notion of message impartiality is quite intuitive, the lack of an objective definition and of a way to measure it directly has largely obstructed scientific examination. In this work we operationalize message impartiality in terms of how discernible the affiliation of its author is, and introduce a methodology for quantifying it automatically. Unlike a supervised machine learning approach, our method can be used in the context of emerging events where impartiality labels are not immediately available. Our framework enables us to study the effects of (im)partiality on social media discussions at scale. We show that this phenomenon is highly consequential, with partial messages being twice more likely to spread than impartial ones, even after controlling for author and topic. By taking this fine-grained approach to polarization, we also provide new insights into the temporal evolution of online discussions centered around major political and sporting events.