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Discourse involves two perspectives: a person's intention in making an utterance and others' perception of that utterance. The misalignment between these perspectives can lead to undesirable outcomes, such as misunderstandings, low productivity and even overt strife. In this work, we present a computational framework for exploring and comparing both perspectives in online public discussions. We combine logged data about public comments on Facebook with a survey of over 16,000 people about their intentions in writing these comments or about their perceptions of comments that others had written. Unlike previous studies of online discussions that have largely relied on third-party labels to quantify properties such as sentiment and subjectivity, our approach also directly captures what the speakers actually intended when writing their comments. In particular, our analysis focuses on judgments of whether a comment is stating a fact or an opinion, since these concepts were shown to be often confused. We show that intentions and perceptions diverge in consequential ways. People are more likely to perceive opinions than to intend them, and linguistic cues that signal how an utterance is intended can differ from those that signal how it will be perceived. Further, this misalignment between intentions and perceptions can be linked to the future health of a conversation: when a comment whose author intended to share a fact is misperceived as sharing an opinion, the subsequent conversation is more likely to derail into uncivil behavior than when the comment is perceived as intended. Altogether, these findings may inform the design of discussion platforms that better promote positive interactions.