In languages such as Spanish, words associated with a negative verb also take a negative form, e.g. No sé nada (Lit. I don’t know nothing). On the other hand, in English this use of double negatives is considered incorrect, and instead only the verb is negative, e.g. I don’t know anything. How does Esperanto handle this situation?
In Esperanto we normally say "Mi scias nenion" for "I don't know anything". So like in standard English, we only use one negation, but we tend to put it into a correlative (table word) if possible, rather than using "ne".
However, it is also possible to use "ne" in combination with a non-negative correlative, i.e. to say "Mi ne scias ion ajn", which is more similar in structure to the English "I don't know anything"
If you use two negations, the overall meaning becomes positive again. For example, "Mi ne scias nenion" means "It's not the case that I know nothing", i.e. I do know something.
Double negatives are not used in Esperanto:
Diru nenion al iu ajn!
Don't say anything to anyone!
However, if you use two negative words, the meaning changes:
Diru nenion al neniu!
Don't say nothing to anyone! (= Say at least something to everyone)
However these kinds of sentences are very uncommon, and I'd rather assume the speaker made a mistake and intended the first meaning.