If the drain is shared with another property AND discharges into a public sewer then the water company is not only responsible for it, they own it and you need their permission to build near or work on it even if it's on your land.
In the case of a Buchan trap they would probably mark it for regular jetting by themselves as opposed to replacing it.
If the drain discharges to a cesspit or septic tank then it belongs ( jointly if applicable ) to whoever discharges into it, whether there is a formal agreement or not.
With properties now having vented stacks the trap is no longer required and with the volumes of water and soilds flushed down nowadays they are no longer suitable, although if there is a cesspit or septic tank that would need a vent.
Is the Cesspit / Septic tank regularly emptied of solids by a tanker company?
If it was mine, I'd trace the drains and then if found to be discharging to Cess / Septic then replace the trap with a pipe.
If you wanted to try and trace it yourself buy 2 different colours of drain dye, put some down from your property and see if it ends up in the pit and then repeat with a different colour at the neighbours, if they'll let you.