All French Property

I would think you need a drainage tile more than a french drain. PVC drainage tiles have holes in them to collect the water and deposit it at the end (near the street in your case) They have a mesh netting that goes over the pipe to protect from dirt filling them up. They are buried in the ground.

Have a look at the flowing link. They seem to have reasonable advice on how to do it.

what sensibleman described is a french drain even if he doesn’t know it your idea of placing along driveway is good and drain should have outlet unless it flows into a deeper drainage well but you should check local ordinances about drain outlets flowing over any sidewalks as this can cause injuries due to someone slipping on wet surfaces

The problem you will have with just trying to install a french drain is that you will have to leave the top exposed so the water can run into the french drain. And, depending on the amount of water you have to deal with, you may have to provide an outlet for the drain, otherwise it will just fill with water and continue to flood your driveway.
Another choice would be to install u-drains along the side of your driveway, routing the outlet at the end to come up out of the ground. You can find these in about 2 foot sections available at Home Depot. They can be connected directly together, or seperated and joined by pieces of pipe.
Another option would be to cut a culvert between your driveway and the neighbors yard to driect the water down the culvert to the street.