Oh wow, this brings back memories:) Neighbors.... the second hardest part of rehabbing properties, behind contractors!
Yes, your situation sounds tricky with it being shared sewer lines! I would defiantly get a lawyer involved to at least get some questions answered and run an extra search for any possible written easement. Do they have a lawyer involved... or is this info all related through a "conversation in the back yard."
What are your plans for the property? Is this a flip? or a rental? or primary? I think that should play a role in the way to handle it. For instance- for us, being a flip...time costs additional money, so I weighed that factor (dragging things through court would cost me even more money). If it was my property to hold for a longer period of time, I might have gone about it differently.
We ended up giving them two options. 1- they can run a new water line on their own land and not deal with us at all, or 2- they can run a new line on our property but in new said location and with a strict easement agreement. In the end they went with a new line and easement on our land. Personally, I don't understand why people would even entertain the idea of running their lines through someone else's land unless it truly was the only way, but they did!
With that, we made sure they paid for all related expenses....the cost to run new line, the repairs to our yard/ new landscaping, our attorney fees, and an extra fee for an easement.... and a deadline for said work to be completed. We had a very clear easement written with a solid maintenance agreement. In the maintenance agreement, they were responsible for paying for any future repairs not only to their physical water line, but also for any damages to our property, regardless of what/who caused the damage. We felt that this would give a future owner some peace of mind. If the new owners wanted to build a 10k patio on it, the neighbor would have to pay to get it fixed if any future leaks /issues occurred.
Being a sewer line ( vs water line), that may be very different though! Older sewer systems were often shared in many ways without easements. I agree with Teresa, talk with the city sewer department. Talk with a lawyer... tell them to have their lawyer talk to your lawyer ( sometimes that calms them down or opens them up to be more rational...?) Getting a separate line even if on your property would be better and defiantly get a strong written agreement to avoid this in the future.
I hope my response is helpful to some degree. I know it is a stressful situation! Do what you need to in order to protect your investment the best way you can. Good luck.