Annoying little beasties, though not fatal to the rose, they merely ruin a lot of the foliage by chewing it full of holes, sometimes to the point of eating it all. Various genus and species within the order Hymenoptera. (This is science-talk for "There are a lot of different kinds.") They are commonly called "Rose Slugs". They are not slugs, however, but the larval form of Sawflies, a wasp-like flying insect.
They are most often host-specific, meaning they go after one type of plant--there is one kind for elms, one for pecans, others for oaks, ash, many other plants, and unfortunately, at least a couple for roses. One type has only one generation per year: you get a few weeks of damage, then they vanish for another year.
This garden seems to have a more annoying type, which produces multiple generations per year, and therefore a lot more damage. I have best luck being alert for initial damage, and rubbing the backs of the affected leaves to kill the tiny, hard-to-see larvae. If I can kill enough of the first generation, damage is significantly reduced. In my less experienced gardening days, the thought of squishing little worms was repugnant, but now that I'm a hardened veteran, I squish away until my fingers are green. What we do for our roses, eh?
Note I mentioned nothing at all about insecticide. I'm not fanatically "organic": in a dire situation I'll break out an insecticide. However, I have yet to face a dire situation with roses in this garden. Even a lot of Rose Slug damage won't kill the plant, and small birds, ladybug larvae, and lacewing larvae do appear to eat the pests. Soon enough the rose grows new foliage that covers the damaged leaves. In weighing the negatives of insecticide vs. the negatives of Rose Slug damage, the insecticide negatives far outweigh the Rose Slug negatives. Hence, green gooey fingers. But they wash off.