I can't tell you what happens behind the scenes at our HS but I can tell you the process our school uses. I have a son at the school and my wife used to be a Dept. Head in the school. Its a large school, about 900 kids per class. Our school has stricter legibility guidelines then the state. Every Thursday a grade report for every athlete is provided to the athletic office. The coaches are notified of any student who is in danger of loosing their weekly legibility. The coaches will notify the student of this and let them know where they are failing. It then becomes up to the student to follow up with the proper teacher to correct their eligibility problem. If they can fix the problem by the end of the day Friday they remain eligible for the next week. If they can not fix it, they become ineligible starting Monday of the next week through Sunday. I believe if they are on the ineligible list 3 times during the season they are dropped from the sport. In addition if you your semester ending grades make you ineligible your are ineligible for the entire next semester. This will carry over from the previous year. So if you are ineligible at the end of the Spring Semester when school resumes in the fall you are ineligible for that semester.
According to my wife most coaches do not get directly involved with correcting problems. The onus of fixing things is on the student.
My kid's high school was similar. Not passing in homework could make a kid ineligible the following week. The coach received a list. He would pull the kid aside and tell him he better see his guidance counselor first thing in the morning. They had until the end of the day the following Monday to fix the problem or they were ineligible that week.
My son graduated in the top twenty in his class. He was on the list three times senior year (my daughter was the good child). It was always for not getting homework in. A lot of the team had a bad case of senioritis that year. It drove the coach crazy. I believe only one kid hit the ineligible list.