Drainage on side of new driveway
I suppose in the US for a normal concrete job, filling around the outside of where the forms were is not part of the concrete contractor's scope of work. This would be done by the home owner who, if he's flush, might hire a landscaping contractor. Where I live people would fill and sod with turfgrass.

I am not a fan of French drains, but prefer surface flow. In Dallas TX I see some people spending a lot on "French drains" to redirect water around slab foundations when in my view the surface should be contoured to do this.

If you sod along the driveway, be sure to avoid over filling. I think you want the level of the soil to be about an inch or so below the surface of the driveway. This includes the soil that comes with the sod. This prevents soil eroding onto the driveway.