The Qur'an and hadith (both Sunni and Shi'i) give the sense that the jinn exist in the same world as us. The Qur'an and hadith describe occasional interactions between humans and jinn on a physical level - for instance, seeing a jinn manifest as an animal (as in the story of the jinn who came to see Imam Ali in the Mosque of Kufa) or hearing their voices audibly. Another example would be the jinn working for the prophet Sulayman (A).
However, apart from those occasional interactions, usually jinn keep to themselves and human beings do not see them.
However, some mystics have held that there is a sort of veil between the human world and the jinn world, which you could call the barzakh. That is, humans and jinn subsist in the same realm, but in something akin to parallel dimensions. I am not aware of any basis for this in Qur'an and hadith, but it could be true.
So do jinn live in the barzakh? To me, the best view is "maybe" and "it depends on what you call the barzakh". If one takes "barzakh" to mean something akin to "mundus imaginalis" - that is, the imaginal realm which humans access through dreams, visions, or imagination, it might be correct to say that jinn live in barzakh. After all, people usually encounter jinn in dreams or the inner realm (for instance, Shaytan whispering to the heart), rather than in physical life. This is similar to how people are more likely to have a dream or vision of a deceased person, such as Imam Husayn (A), rather than physically seeing them walking down the street.
If one takes barzakh to mean specifically the place where human spirits go after death but before the Resurrection... while this meaning of "barzakh" may have some relationship to the above, I am not aware of any texts which specifically say that deceased human spirits live in the same place as living/deceased jinn, although I don't think we have any texts that rule out any sort of interaction ever between living/deceased jinn and deceased humans.
In any case, it seems as if the existence of the human being after death is more similar to the jinn, because the human being in the barzakh is less encumbered by the physical body and able to move around more.
In fact, I don't think we have any texts that say what happens to jinn after they die but before the Resurrection - for instance, do they go to their own barzakh? - however, the Qur'an says that, after Resurrection and Judgment, the jinn who go to Hell and the humans who go to Hell are all mixed together in the same Hell. It does not specify whether the jinn who go to Heaven and the humans who go to Heaven go to the same Heaven.
Anyway, it can be quite difficult to make concrete statements about spiritual cosmology, metaphysical realities, or other things which are not tangible. Even if someone says something, it is really quite difficult to prove whether it is true or false, except in the case of Qur'an and accurate hadith, which we accept on the basis of them being divinely inspired or approved.
So, this, in short, is my understanding of what the Qur'an and hadith say, and also what some mystically inclined thinkers have said. God knows best!