This is a good question, and while I can’t point to any particular texts that answer it exactly, I go back to the Buddha’s admonition to his Sangha prior to his passing (they were lamenting “who will be our teacher?”), which is to look to the Dhamma as the teacher. Thus, the importance of studying the EBTs and having a meditation practice. Then, with this foundation, we have some clear sense of which among the lay teachers is actually teaching well grounded Dhamma, and which are promoting something else.
In some other traditions, there is the guru-centric model, and the models where some teachers perform a “transmission” of teaching authority to their chosen students. These models have, as we have seen, created a number of problems and issues in western Buddhisms.
Within the Thai Forest tradition, I’ve not heard of Stream entry as being a necessary qualification for teaching. Again, it seems to me the idea is to cultivate within oneself a measure of knowledge and wisdom, and then venture forth to see which monastics or lay teachers have the kind of knowledge, experience,scholarship, and passion for teaching such that they distinguish themselves in the field. I know that within the Forest tradition it’s not unusual for abbots to send their students away to other wats, so that the student gets a broad training from other perspectives. In the wat where I ordained (anagarika, samanera temp.), one of my colleagues who fully ordained has been sent all over Chiang Mai region to train at other wats, and then, time to time, comes back to the “home wat” to teach and train. The abbot acts not as his “guru” or root teacher, but as a mentor, coach, or supervisor of his training.