That sounds like a Part 135 scheduled op's FAR to me, not 141 training, but I don't know the 141 rules either
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Thats no rule in 141 either. I currently have 5 hard students, and I wouldn't mind a couple more. Here, they assign us some students, but we can request more. It takes forever to get more, however. We can pick up students on the fly as well (Their instructor is sick, or on vacation, or they haven't been assigned an instructor yet, etc.). I'm enjoying the job, its more mentally stimulating that flying divers, and less actual work (I love flying, but 7.8 hours a day in a 182 in 90 degree weather, climbing at Vy to 11K over and over got tiring after a while).