Not really a mystery. Although few individuals at the time could afford to own one, some could and did. More importantly, there was no requirement to property mark them. Many purchased in small lots by sporting clubs, businesses with company shooting teams, schools, or even smaller SA units were never given any such marks but were still used by numerous people for marksmanship training. The SS even required their members to qualify with the .22 trainers, but how many SS-marked trainers do you see? And check out period photos of the trainers in use, and note how many rifles show no property marks. There are photos of military instructors directing training for youth groups and civilians, but military purchases or use of trainers was almost non-existent, or you would see WaA acceptance stamps to show that. A 30-year-old draftee might well have had some small bore training prior to induction, but not always.
Steve