It's obvious you are a very loving & responsible doggie owner and that's wonderful! I have 2 dogs that play in the yard everyday and only recently learned that my aloe vera, Azaleas, Hostas & cyclamin, Chrysanthemums & Ivy are toxic to dogs, yet I've had them in the garden for quite some time. My next door neighbor has an avocado tree that hangs over into my yard and it has never been an issue. My biggest worry mushrooms! But as far as a screen~ I'll chime in with my suggestion~ Dodonaea viscosa cv. purpurea (Purple Dodonaea). You can almost plant it and forget about it. They are evergreen, very drought tolerant and fast growers and do well in our zone. They can get really tall & wide but not invasive at all. They need to be kept pruned (about once a year or every other year)for desired size & shape and to look their best when they get older, but are super easy to prune since the branches are spread out. A few cuts and your done. I think it would be nice for a cottage style as well because they are very informal looking. Just make sure you don't plant them in an area where you have other plants that require a lot of water because over watering will kill it. I have one in a long deep planter that's built in to my deck (approx. 3'deep x 8' long x 12" depth) and give it about 2-3 feet around the roots where I don't water as frequently as I do for plants at the other end of the planter. If you dont like the purple, there is also the green variety. Here is a link that might be useful:

Cherry Laurels love shade and grow fast once established (evergreen shrubs to small trees) but are poisonous to dogs. I planted a row of them on my north side to block out a purple house on the hill, and will just take my chances that I don't get a stupid dog when I get another dog :) Camellias also love shade and they flower in the fall and winter. They are shrubs but really they are small trees, they do grow slow however. Tulip poplar would grow very quickly to reach for the light, and the leaves would get rather large in the shade. That tree gets big though! Also, a thought: drive around and look for houses in similar conditions... What is growing in those situations?! I “cheat" all the time this way with landscaping. Also, maybe you could befriend the new owner and convince them to grow cross vine or Virginia creeper on the side of their house? Think understory trees too, several people have mentioned some great ones (especially hornbeam/ironwood/musclewood, dogwood, and redbud). I would add serviceberry and beech to that list (although beech eventually gets large. It keeps its leaves in the winter).

I feel your pain. I live near the ocean in a tract of smaller single family homes that are all one story. Our neighbor behind us built into their entire backyard and then added a huge second story with a balcony off their upstairs master bedroom that looks right down on our home and yard. Their purpose was to have a (very distant) ocean view, but they completely blocked our very nice mountain view and took away our privacy. The lack of privacy bothered me the most. We planted bamboo at the fence line and have been waiting for it to fill in. It is not perfect (or for everyone) but each year we get a little privacy back. We did our research first and built in a barrier to keep it from going everywhere. We also are proactive and maintain it to make sure it stays in its lane, because the barrier isn't perfect. I think grape vines or bougainvillea would also give you some privacy, be easier to maintain, and grow nicely in your area.

We are in zone 9b, Riverside County, CA near Temecula and we have 4 blood oranges in the front yard. They are beautiful all year and would provide a screen, but our yard is huge, and they are far apart. On the side we have a Manila mango with amazing growth. Macadamias grow large and we have the Beaumont variety too, along with Bearss lime and a couple regular oranges and a couple lemons. One lemon is enough, but someone threw a seed down and a random lemon grew. Those lemons are tiny and I love them. They are culinary (not sweet) so every night I squeeze one over an avocado, and put it in a green salad with oil, salt and pepper for dinner. Lemon rinds and juice will freeze also. If you want height, get the standard size citrus. Anything fruiting will produce fruit, and at some point your neighbors won't want any more and you will say "STOP!" but they don't. :-) I LOVE fig trees but they go dormant in winter. Awesome shade in summer. Before you buy check the watering and fertilization requirements. Mangos LOVE water!