We discovered recently that our party wall does not align with the dividing fence so party wall is on our side of the fence and garden. But neighbours would still share responsibility for it.
Sounds like the same situation with your chimney.
I have worked on houses ,but older ones , where the flue from one house goes partly into next door. Very unpredictable. If the fireplace is still there, a lit piece of paper will show smoke out of the pot, which is your pot
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