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As far as I know: if it's within their boundary it's not a party wall, so you shouldn't touch it, including paint. They have rights to access for maintenance, but not for building new.
If they were building new I think they'd need a party wall agreement anyway for the footings, not sure on the technicalities of whether this then classes the above ground section as a party wall.
If it straddles the boundary, then it's a party wall. As far as I know you don't need a PWA for painting and drilling an interior PW, but I don't know about exterior, where it could have ramifications for weather proofing etc. I think there's a whole forum somewhere dedicated to this stuff. A PWA or an access for maintenance contract will have a time window attached which could be used to minimise disruption.
I was wondering about party walls facing onto other properties and stuff and who can do what to them the other day.
Taking this as an example:
if you were next door to it would you be allowed to paint it, screw things in so you could put a trellis or shelves or something up and that kind of jazz or does it belong 100% to the person who has the extension?
Also, how does building such a thing work? I assume they need a fair bit of access to do that kind of work. Is there any right to dictate the terms of that access or is it like the party wall act where you can delay stuff but realistically it's going to happen. I assume you wouldn't want someone building a big extension in your garden in the middle of summer.