Is it normal for my driveway to get dark blotchy spots after it's hosed off?

A blower by itself won't stain concrete, but what it's moving absolutely can. Wet grass clippings and mulch gunk blasted onto a driveway will leave exactly what you're describing, random dark blotches where the wet organic stuff sat on the surface for a while. Grass and mulch are full of tannins, same stuff that stains a driveway brown under a pile of leaves, and a blower just spreads it around in an uneven pattern instead of a neat shape. That's why it looks random across a few panels and why dish soap barely touched it. Tannin stains soak into the pores of the concrete, so the rain rolling off did nothing either.

The other possibility is drift from whatever they sprayed next door. Most weed sprays don't really discolor concrete though, they just dry invisibly. Fertilizer blends with iron in them can leave dark or rusty blotches, so it's worth asking the neighbor what the crew put down that day. If it was a spray, that probably wasn't it. If it was a granular feed, that's a real suspect. Either way, don't panic about the surface. What worked for me on a similar mess was an oxygenated cleaner, the powder kind you mix with water, scrubbed in with the same stiff brush and left to sit for 15 or 20 minutes before rinsing. You may need two rounds. It lifts organic staining without the aggression of a pressure washer, which matters on older concrete since a tight nozzle can absolutely eat the surface and leave you with permanent light stripes worse than the stains. Honestly, a lot of this type of staining also fades on its own over a couple months of sun and foot traffic. If you can live with it for a bit, that's a valid option. But if you want it gone sooner, try the oxygen soak on one panel first as a test, and only move to a pressure wash on a wide fan, low pressure setting if that fails.

First time posting here so sorry if this is a dumb question, but does the blotching get worse when the concrete is wet? I ask because I had something similar on my front walk last summer and it turned out to be where my sprinkler was hitting the side of my garbage bin, water sitting there and dragging gunk down onto the slab. Once I moved the bin the marks slowly faded over the season on thier own. Not saying that's what happened here, just that it took way longer than a week to see any difference, so maybe don't judge it too fast. Did the patches look darker when you hosed the spot you scrubbed?

Been chasing driveway stains since before the oxygen cleaners were even on the shelf, and here's an old trick that still works. Get a box of TSP from the hardware store, mix a scoop in a bucket of hot water, and scrub it with a stiff push broom instead of a hand brush. Covers more ground and the broom bristles dig in better. Let it sit a few minutes and hose it off. That's how we used to strip tannin and leaf stains off garage floors, works just as well on blotchy slabs like yours. Try one panel before doing the whole drive so you know how it compares to the rest.

Before you scrub anything else, I'd just knock on the neighbor's door and ask what the crew was doing that day. Mine did basicly the same thing to me a couple summers back, turned out they'd fertilized with something that had iron in it and the overspray left rust-colored blotches all down my edge. I called the company and they sent a guy back with a cleaner and it mostly came out. Way easier than DIY-ing it and guessing. Worst case they say no and you're no worse off than you are now.

Update: tried the TSP in hot water with a push broom like one of you suggested and it actually worked on most of the blotches, way better than the dish soap did. But the marks along the edge facing the neighbor's lawn are still sitting there and they do look rust-colored, so now I'm wondering about that fertilizer overspray theory. Does iron staining fade on its own or does it need a dedicated rust remover?