At one barn I was at, the outdoor was right on a busy road, and they had planted arborvitae between the arena and the road. A couple had died or broken off, so you'd have a pretty good screen, then a gap, more screen, another gap...add in the horses hearing and not seeing vehicles and it was a pretty scary ring. We never used it much as the footing was terrible, but I almost think having no screen would have been better.
If your climate will support it, bamboo is da bomb! Expensive initially, but grows like stink! I blocked out a neighbors barking Rotweilers and ugly two story house instantly. The dogs are still within 30' of the arena, but they can't see us! They need watering for the first month, then nothing. Ever!
It does wave in the breeze, and new horses look askance at it. Good practice for concentration. Mine are very close to the arena--about 15'.
It will also go EVERYWHERE. It is easy to grow, but it is invasive and hard to control. But it does do a great job of hiding the trailer park around the corner from me. All year.
I feel your pain OP... I have a ring about 10 foot from the road and I'm in farm and Amish country. All manner of stuff goes by on the road. Cows, ATVs, monster tractors, buggies, goats, motorized wheelchairs, kids on bikes, draft horses pulling huge carts with foals running alongside their mamma's. We own the only donkey in the neighborhood so I guess we are safe there
But I don't have anything between the road and the ring and I plan to keep in that way. I think it would be worse if they could hear things and not see them. What I notice is that the horses usually just take their cue from me. If I'm focused on what I am doing, neither of us pay any attention to whats on the road. Of course the occasional green horse gets distracted, but generally they all get used to all the stuff and don't pay any mind to it. So I understand what you mean by it bothering you more than the horses. My advice would be to practice closing out the outside stuff and getting "in a bubble" with just you and your horse.
And goodhors - I ride english - love my arena right where it is.
I ride occasionally at a lovely arena that is along a quiet road, and between the arena and the road is a thick, impenetrable wall of shrubs, trees, etc. Across the road is a property with a house quite close to the road. They bark and fuss at each other a lot, shoot targets, slam car doors, and slam the front door. If I could see that they were out there, I would know, and my horse would know, oh it's going to be noisy now. As it is, we don't know. So we both jump when they just start BLAM BLAM BLAM-ing.
I sympathize with the OP about wanting the screen more for "me", than for horse - I have one neighbor that has caused me endless hassles & we are no longer on speaking terms - when I ride he seems to delight in coming outside, standing in the back yard & staring at me. Of course I *should* get over this (and I'm working on it) but I've decided I'm putting up privacy fence on that one end this spring (then he'll have to get on a ladder with binoculars to annoy me, but at least then he'll obviously be the peeping tom fool!)
I have fairly dense woods on one end, an open 2 acre field with a few houses over there, the sacrifice paddocks on the other, and then this opening to my neighbor from hell that I plan to shield off with privacy fence, then plant evergreens.
I can choose to school either in the enclosed arena or in the 2 acre field, so it works fairly well for us.
I know they need to see & get exposed to things, but for my sake I want to keep it fairly quiet & private as much as I can at home. I take them out plenty to see what the rest of the world has to offer before I show - at home I want to relax! Being an introvert, that means privacy.
Anyway, do any kind of evergreen grow fast for you? In my area, White Pines & Douglas Fir thrive and if planted at 4-6', make an excellent long lived screen in 2-3 years.