How do sub-surface drains (e.g. French drains) work?
A French drain doesn't move the water anywhere it just creates a void that can be filled with water for the short term. You could do the same thing by burying a barrel with holes in it and landscape fabric around it. Eventually it would get full of water and if the rain (or garden watering) doesn't stop it will still flood the area.

Nothing prevents the water from falling back out of the pipe except hydraulic pressure. But that is the idea. As the ground dries back out the water leeches back into the soil. There is not usually an ultimate destination but you could create one by connecting solid pipe to the perforated pipe running downhill to another area. 

You can certainly do this on your side of the property to help reduce flooding.