I can tell you something about ancient Greek and Roman sources; other commenters may be able to offer information for other historical contexts.

The short answer is that we never really know for sure how accurately the quotations given in textual sources reflect the things people actually said.

It was an accepted custom among ancient historians that they did not try to record speeches word for word. Rather they gave an impression in their own words of what points and arguments a speaker made and the general flow of debate or conversation. People who gave public speeches (whether in political contexts, in a court, or in some other setting) sometimes also published a written version of their speech, but these written versions were often polished up and rewritten before being released.

We rarely have multiple different accounts of the same speech or phrase to compare against one another, and when we do they don't always match up. When they do agree with one another, it's hard to know whether that's a sign that they are accurately reporting the words spoken or simply that one source was copying from the other (or both from some third source, which may be lost to us now).

Given all of the above, historians of the ancient Mediterranean don't often get into questions of how accurately our surviving texts reflect the things people actually said, since it's usually impossible to arrive at a satisfying answer. When we do consider this issue, though, some of the questions we ask are:

1. What kind of setting was this quotation supposedly given in, and how many people were present? The more public the setting and the more people there to hear, the more likely that someone accurately remembered and wrote down the words.
2. How soon after the event was the source written? The sooner the words were written down, the more likely it is that they were accurately remembered.
3. If the surviving textual sources do not come from soon after the event, is there reason to believe they are based on earlier written sources that are now lost? If so, how much confidence can we have in the intervening sources' accuracy?
4. How long is the quotation? The longer the speech, the less likely it is that the person who wrote it down remembered it precisely. On the other had, a short, pithy phrase with a lot of punch may well have been something a writer thought up after the fact rather than what was actually said in the moment.
5. Was it in anyone's interest to misrepresent what was said? Do our surviving texts come from people who were hostile to the speaker, or who had a vested political, social, or economic interest in a famous person having endorsed a certain attitude or argument? If not, are they likely drawing on earlier sources that did?
6. Who was the text intended for? A text intended for wide publication among people not present for the event might be more likely to misrepresent a person's words--whether positively or negatively--than a private letter between friends, especially if the speaker themselves was part of the writer's social circle and might see the letter.