In English, there should be only one use of "not" with a verb. In the second one, there are two, because "don't" = "do not." This is called a "double negative," and it is not grammatically correct in English. The same is true when not is used with other words that are negative. For example, "I don't never do that" is incorrect because of the double negative.

In english we don't say "we don't not necessarily", it's grammatically incorrect. You either do necessarily or don't necessarily need to....whatever. There is some intonation differences between "don't" and "do not" that you should look up but nothing specific to the phrase "don't necessarily".

In English, there should be only one use of "not" with a verb. In the second one, there are two, because "don't" = "do not." This is called a "double negative," and it is not grammatically correct in English. The same is true when not is used with other words that are negative. For example, "I don't never do that" is incorrect because of the double negative.