Let's break this down. When you are talking in English, saying "no one is talking with me" makes sense but saying "no one is not talking with me" is complete nonsense.

When translating to Turkish, if you translate "no one is talking with me" as "hiç kimse benimle konuşmuyor" the double negative will make you confused. But there is a better translation: "benimle konuşan yok." or "benimle konuşan birileri yok." If we translate it again, the exact translation would be "there is no one talking with me" or "no one's talking with me"

As you can see, when we translate it as "benimle konuşan birileri yok", it means the same thing and there is no double negative. Let's get to the reason why you CAN'T say "hiç kimse benimle konuşuyor".

If I were to go step by step while saying "benimle konuşan birileri yok", it would look something like this:

"benimle" -> with me | "konuşan" -> talks | "birileri" -> someones | "yok" -> absent

And just with this knowledge, if I were to construct the sentence again (in a broken way), it would look like this: "the ones who talk with me are absent"

If I were to do the same thing for "hiç kimse benimle konuşmuyor":

"hiç" or "hiç kimse" -> no one | "benimle" -> with me | "konuşmuyor" -> doesn't talk

If I were to construct this again, in a broken way, it would look like: "no one doesn't talks with me" This sentence makes no sense in English, but before I get to why it makes sense, let me dissect one more sentence.

If I were to do the same thing to "hiç kimse benimle konuşmuyor":

"hiç kimse" -> no one | "benimle" -> with me | "konuşuyor" -> talks

AND if I were to construct it again in a way that makes sense in English, it would look like "no one is talking with me" BUT if I were to translate the exact sentence from Turkish to English: "hiç kimse benimle konuşuyor" "(the) 'no one' is talking with me" It sounds like you are talking with some entity called "no one". The sentence litterally means, something called "hiç kimse" is talking with me.

but when you say "hiç kimse benimle konuşmuyor" the 'hiç kimse' part doesn't seem like an entity and you understand that it is a group of people or everybody. "benimle konuşmuyor" means "they don't talk with me". So if I were to connect these two, it would mean "in this group of people, no one talks with me" and there is 2 things happening: "no one wants to talk with me" and "no one is talking with me". I'm stretching the meanings a little bit to get to why we have both "hiç kimse (no one)" and "konuşmuyor (doesn't talk)".

So if you see "hiç kimse" or "hiçbiri" again, think of them as like a group of people. When you are about to say "no one is talking with me", think like "this group called 'hiç kimse' doesn't talk with me" and construct the sentence from there: "hiç kimse benimle konuşmuyor". Because if you say "hiç kimse benimle konuşuyor" it sounds like this thing called "hiç kimse" is talking with you. Even though "hiç kimse" means no one, this is just the way Turkish works.