In English "I don't really know" means: "Actually/in reality/if I think about it further... I don't know", where 'really' modifies the phrase "I don't know" to make it milder/more casual (as opposed to being blunt), or modifies the verb 'to know' to make it clear that the speaker doesn't fully (really) know whatever's being spoken of;
or, if really is stressed: "I was only joking - I don't really know...".
In contrast, with "I really don't know", 'really' applies more to the negative don't, so it means: "I certainly/definitely do not know", perhaps suggesting that you've tried to know, but you still don't, thus you're sure and you "really don't know".
Similarly, could someone explain how the word placement in the Italian sentence affects meaning?