We recently replaced the concrete on the side of our house as there had been lots of pooling. The new concrete (2nd pic) is pitched higher than the old concrete (1st pic). This angered our neighbor and he claimed that all the water was now draining to his side (I've yet to see any proof of this). After some arguing about property lines, the neighbor started to build a French drain on his side. Our side is the concrete and the narrow patch of dirt up to the plastic barrier in the 2nd pic. I had assumed that the property line ended at the concrete, which was not the case as I later confirmed. I'm not sure if I should build my own French drain or leave as is.

In the pic with the old concrete, the dirt was higher and angled down towards our house. There's about 24 inches between the over hangs of our roofs. We haven't had any flooding issues yet.

Any help is much appreciated!