When I have done this, there was no cost. There was a couple day leadtime though.
They located the utility type lines; water, electric, cable, phone from the street to thier connect point at the house. They did not locate sprinkler lines, electrical out to the shed, leach lines, etc. That may be the third party...
We did work at a place where we had USA(Underground Service Alert) mark the utilities. The owner was having a pool put in, so we measured to where the pool lines were, marked them on our plans, and took pictures.
Everything was fine where the USA marking(utilities) were; I trenched up to them, then hand dug around them.
The pool though..... After we had measured and marked where their lines were, referenced pictures etc, I still dinged a pool line with a jack hammer. Had to half drain a brand new pool to fix the line. I was not a happy camper...
Turned out the pool company moved a couple lines. After, we had met with them, measured, and took pictures specifically so this issue would not happen.....
All of the other lines have been marked, so I did in fact call the city and turns out that a third party vendor marked those lines. You all may be right, I just may have to have it done myself and just add the cost to her bill.