In debaterland terms, some quick points on both sides:
Against:
*Freedom of speech can be harmful to individuals, in that it basically allows anyone to say anything, no matter how offensive or derogatory it is. This can threaten people's sense of self-worth, and especially in some cases (such as when one criticises an entire culture as immoral, for instance), it can be existentially damaging.
*More broadly, freedom of speech can generate wide societal harms as well, when we allow hate speech and that sort of thing. Often, well-deployed rhetoric with immoral content can lead to widespread immoral behaviour (as people will come to believe the content of said speech, and society's culture will gradually re-orient to accept it) - really lame example, but case in point, Nazi Germany and the slandering of Jewish people.
*By the same token, when individuals are given freedom of speech, they may say things which can (reasonably preditably) incite violent response. Example - film depicting the prophet Muhammad recently.
*Freedom of speech can detract from legitimate intellectual/rational discussion, in some cases, when people are allowed to say inflammatory things. For instance, on AN (yes, I'm using this as an example), a conversation might start off as reasonable, but often, as soon as somebody begins to troll or flame people, the on-topic content becomes lost in a sea of emotive/aggressive responses instead; this sort of thing is often inevitable as well, as people tend to respond much more strongly to emotional stimuli than rational stuff. This is often justification for why we ban trolling/flaming - because it detracts from more important dialogue.
In favour:
*The classic John Milton/John Stuart Mill free speech defence - that we need people to say the wrong thing so that we can, as a society, work out what is rationally right. Argument often only progresses further or makes sense when there is a counter-opinion put forward - without freedom of speech, society doesn't truly understand why something is legitimate, because we don't have opinions that are outright wrong (such as "Jewish people are all evil") to contrast against opinions/ideas that are good.
*Not only that, but often hate speech is necessary for legitimate dialogue to come forward, not only because it contrasts against good ideas, but because it actually provokes such discussion to begin with. For instance, Waleed Ali only publishes papers defending the legitimacy of Islam BECAUSE so much anti-Islamic sentiment comes out - he is writing because it has become an issue through the hate speech, not just because he has seen the hate speech and has thought of the opposite argument in the process.
*Slippery slope argument - without lobbying for freedom of speech, we implicitly grant governments the ability to censor things without our consent. Sure, we can still lobby against them censoring the internet, etc., but it weakens our case if we have to justify this against the fact that so many other things are censored.
*Harms-based argument - more active harms actually occur when we don't have freedom of speech than when we do, regarding discriminatory action and the like. With freedom of speech, for instance, people can voice their bad opinions, and can be publicly shut down for them. Without it, these people either hold their opinions and are never told off for it (and thus never understand why they are wrong), OR get told they can't say something and grow resentful, further cementing their illegitimate beliefs.
Check out "Areopagitica" and "On Liberty" - classic texts for freedom of speech.
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