First, you need to understand some basic property ownership concepts
Here are two images showing the chimney between the two terraced properties. Yours is on the left in blue, and the neighbours on the right in red.
Everything in red is owned by the neighbour - so he has half the party wall, the roof above his property to the centre of the party wall, and the flashing to his roof.
Everything in blue is yours, and you can see how the chimney goes over onto his half of the party wall as it leaves the roof. This will be the boundary also. So yes, the chimney is owned by you.
The leak which was described earlier (by the EHO) is typical of a leak from flashing. I can't be 100% certain having not seen it, but that's what I am going to presume. And its what you are entitled to assume until proven otherwise [by the neighbour] via a full and detailed inspection
As such, the leak will be via defective flashing. This flashing on the neighbours side of the chimney will be the neighbours responsibility as it is flashing to his roof. In legal terms, you have given him the right to install flashing to your chimney to seal his roof
So that's the gist of ownership and responsibility
Just a quick mention about the EHO's letter. Essentially it is meaningless. The EHO has made some bold claims about the source of "a leak". Firstly, chimneys don't leak water. They are not water containers so can't leak. This is not being pedantic, but a fundamental issue when referring to claims of Statutory Nuisance (EPA 1990) and the like.
In addition, unless he has been up on the roof to inspect it, he can not, and should not have made the definitive claim of stating where the leak was coming from. Does he know that there are no broken tiles, slipped ridge tile or defective flashing? No
Finally, an EHO is not professionally qualified to comment on building structures - unless he also has a qualification in this field
So bear these concepts in mind, and I'll email or PM you a link to a drafted response. If you want an email, then you'll need to PM me an address or email my address in my profile, or profile comments