Assigning Teacher Recs = Same Rec for all Schools?

So after a teacher has been invited via email to fill out a rec on the Common App, is that the only time he/she will have to submit a rec? After that point, is it left to the student to assign the teacher to other schools?

To put the questions in context:

I’ve had a teacher submit a rec to Brown, who received an email request from the Common App after I invited her. No surprises there. However, after I “assigned” her to Oberlin so she could upload a rec to that institution, she said she received no email. Does that mean the rec she used for Brown will be used for any school I apply to (AKA, a teacher only has to submit a rec once, and it will be used for all schools the student assigns the teacher to)?

As is probably obvious, I’m incredibly confused and will be beyond thankful for any clarification!!