Jack has covered most of it. IF you were to 'divert' water (to add to what Jack has already said), where it was to start flowing BACK onto a neighbors property, you MAY be sued to have this 'new problem' corrected (and you'll be right back where you started.) FWIW, in my jurisdiction, since the homes were 'tiered' when they were built, the water from the UPHILL neighbor always flows to the downhill neighbor, and so on. If I was to do something to try and KEEP the water out of my yard (and in my neighbor's), I CAN be sued. So, what are you left with???
You're left with either piping it to the street (which should be OK since it's considered storm drainage.) Or, building a dry well. As Jack already pointed out, your drywell may actuall 'attract' more water than you want. IF your soil is naturally wet, then your drywell is always gonna have water in it. Soooo, if this is the case, when it rains, the well will be be 'pretty full' to begin with. IF, on the other hand, your soil is fairly dry, you'll have to do some figuring on how big to make your well. When and if you dig one, be sure to use Filter Fabric (not landscape fabric) and 3/4" stone. If you make it big enough, I've seen them slightly larger in diameter than an average Gazzebo where the water 'runs' to the perimeter of the gazzebo and drains 'downwards'. Of course, the Gazzebo 'gravel' needs to be a little ABOVE grade so that if there's any standing water from a heavy rain, your Gazzebo isn't sitting in a pool.