Do I need a party wall act?

No just a chat with your neighbour should be suffice, But a shared chimney should be treated as a single unit & any work required should be carried out on the whole chimney as any defects on your neighbours side can still cause dampness on your side & vice versa water will always take the easiest route to escape.

Repointing only your side of a shared chimney stack would normally be classed as routine maintenance and may not require a Party Wall Notice, provided the work does not affect the structure or your neighbour’s side. However, because the soil pipes share the same flashing, removing one could disturb a shared weatherproof junction. I would recommend notifying your neighbour and having the proposed work inspected first. If any shared brickwork, flashing or structure must be cut into or altered, advice from a party-wall surveyor should be obtained before work begins. A competent roofer should be able to inspect the arrangement and confirm the correct approach. Dean