Water Drainage Towards Neighbor's House
Actually, I think we do get a good idea of the situation and you are correct to worry about (and be considerate) to your neighbors. 

You can't CHANGE anything that would "harm" your neighbor's property, but you can "improve" conditions on your property. So, if a tree branch extends over the fence, you can cut it off. However, you can't deform the tree to the point that you kill it. Likewise, you can improve the drainage on your property, but by doing so, you can't harm your neighbor's property, (I.e.: create a drainage swale on their property, soak their land creating a swamp, etc.) 

So, installing a french drain is logical and could solve a lot of problems for you, you just need to be careful: 1) how much water you gather, and 2) where the french drain ends, and 3) how it ends. 

If it's surface water you are collecting, you might consider:  1) diverting it before it gets to your property...up at the street so it doesn't run down your driveway, and 2) collecting it in a french drain system and letting it disperse out of your french drain system well before it gets to your neighbor's property. I'll discuss how to do that later. 

If it's subterranean water, (soggy ground, even when it hasn't been raining for awhile,) then collecting could be a problem. Currently the water is in the aquifer (high water table) and modifying the aquifer is nearly impossible. I've only seen one time when a landowner was able to lower the water table by installing a french drain to divert some of the water closest to their house and discharge it further downhill. 

A french drain to me means a trench filled with gravel with a perforated pipe in it to transport the water. So, we use french drains for two purposes: 1) to gather and transport water where we want it, and 2) to gather and store water...yes, store water. When we store water, we excavate a trench (usually 2' wide because the backhoe shovel is 2' wide) and 2'-3' deep. Then we fill the trench with large round rock with no fines (like: 3" minus to 1".) This creates a lot of voids to store the water. You can line the trench with filter fabric to keep the soil from washing into the voids. Most people don't know that about half the water will dissipate through the soil (depending on the percolation of the soil) and about half will evaporate. Evaporation occurs when the wind blows across the ground and draws the moisture out...and I'll bet you have wind in the northwest. 

Now, the overall design: So, no matter what you're picking up: surface or subterranean water, you want to keep it contained in your new french drain system for as long as possible. To do this, you want to install each leg (and especially the last most downhill leg) as flat as possible. Remember, you're not transporting it to be discharged, because you'll wash your neighbor's land away, you're gathering it to store until it can be dispersed. Also, make sure the top of the trench is level so excess water will "sheet" out of it, not "stream" out of it. 

By the way, the only reason to use a 3-4" perf pipe in the trench is to EVENLY DISPERSE the water. If all the water lays near the top of your property, it won't dissipate as quickly. 

So run the french drains where you need to collect the water, including roof drains, but then extend them across the slope keeping them as level as possible. Normally we don't recommend putting surface water (downspouts, etc.) with subterranean water, but I doubt if you have sufficient land to provide two systems and it all has to be dispersed on your property anyway.

Below the last most downhill leg of your french drain, you can landscape with water tolerant plants (talk to your local nursery) and provide ground cover with a 2-3" layer of decorative rocks instead of grass for the excess water to disperse into and eventually evaporate. 

Note: It's probably best to start at the bottom of the slope and work your way uphill. Otherwise, you'll be collecting water and it will want to run out where you're working. Also, a hydrologist or a civil engineer with a hydrologic background can calculate the size, depth of trench, etc. for storm water conditions in your area or start at bottom and see how it goes...