Do you say "别担心” or ”不担心” for "don't worry"? Or something else?
别担心 is equivalent to don't worry, but with a little twist. You can only use 别担心 when you are talking to someone and addressing him/her at that moment; While 你别担心啦 is correct, 我别担心，他别担心，他们别担心 are all incorrect because 别 here is a soft command word which means 不要. Of course, you can't command yourself or themselves to not do anything. You can only command the one that you are currently talking to, which is 你。你别要想我。
不担心 is easier; It simply means don't worry, there are no twist here. You can use it freely. So 我不担心，你不担心，他不担心 are all correct. 不 simply means No. No worry = don't worry.
别+[verb] = don't+ [verb]. It is either a command or request
不+[verb] = not + [verb] . It indicate negative
You can say '我不担心'(I am not worry), '你不担心'(you are not worry) and '他不担心'(he is not worry) -- Opposite of '我担心'(I am worry), '你担心'(you are worry) and '他担心'(he is worry)
You can also say '你别担心'(don't worry) But you cannot say '我别担心' or '他别担心' because command or request is always directed to second person.
However, you can say "我叫他别担心" in which '别担心' is a a quote from you directed toward a second person at the time.