I had a similar situation but with a council property. They used one of their own guys, I employed my own. No option to share. It all went smoothly enough once I'd pinned down who I should be dealing with. Slightly more expensive but not hugely in the overall scheme of things.
Just looking, the council charged me £450 + VAT for the party wall agreement. (The other side cost me more than double that as they went with a private firm.)
I had a similar situation but with a council property. They used one of their own guys, I employed my own. No option to share. It all went smoothly enough once I'd pinned down who I should be dealing with. Slightly more expensive but not hugely in the overall scheme of things.