You don't need a solicitor to do it, you can get a template and fill it in yourself. this site is useful. You do need to nominate a surveyor.
You hand them the paperwork in advance (2 months I think). If they have issues, you go into dispute, but that doesn't necessarily have to be a big deal. On the whole as a process it's normally a bit of a technicality,
We are doing this at the moment. What exactly do you have to serve the notice for - are you putting a beam into an adjoining wall or digging within 3 metres of their property? You can get started on other parts of the work that don't fall under the party wall agreement.