Mentor/student responsibility splits are team specific and even within a team may vary year-to-year as experienced mentors/students move on and the inexperienced fill their shoes.
If you just want specifics for our team here are our lead mentor duties
Advisor Handbook or for the run-of-the-mill mentor:
Mentor Handbook, our corresponding student officer duties
Officer Handbook or generic student responsibilities:
Student Handbook. Fundraising, travel, etc. have been taken over by our parent Booster Club:
Parent Handbook, but responsibilities beyond requiring accounts and contracts to be through legal adults are shared with students.
Generally, duties are split based on interest with the thankless jobs ending up most often in the hands of whoever blinks first. Core duties are the ultimate responsibility of the lead mentor with delegation to students, mentors, and parents.
Other than that I'm going to take a different tack here just in case...
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Has your teacher declined to assume some responsibilites that you all automatically expected them to assume?
We all must form a true appreciation for what teacher's volunteer for (and sacrifice) as opposed to what they are required to do by appointment by a school board of education, or through the goodwill of their hearts. People will do more for you if you don't take them for granted.
From a purely official standpoint only the advisors/mentors appointed to run the team by the school or chartering organization have specific official responsibilities. Those responsibilities address the administration's concerns and are typically legal of course. Lawsuits and the Board of Education calls only the Lead advisor/mentor to task, not the students. Official responsibilities usually include: hold meetings covering a specified number of hours, insure student safety, and process paperwork.
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Everything else is at the courtesy of the teacher, so you should be especially nice to them since they are usually volunteering their personal time and money from there on out. They usually accept and take on additional responsibilities, but those additional duties are on a purely volunteer basis. Usually safety training is assumed, however, with a club a teacher's official responsibilities may only include a requirement to be present at meetings and the basic safety/welfare of the club students.
Chaperoning the team, teaching the students, spearheading fund raising efforts, corporate relations, public relations are strictly speaking all additional volunteer efforts that can be spread over many other helping hands. Typically, teachers do take on much of these duties through a personal sense of responsibility and a wish to see the team succeed, however, teams must understand and be properly appreciative that it is indeed a personal volunteer effort and is not required. Some teachers just may not be able to do all these things due to overloaded commitments, while another teacher may go way above and beyond their responsibilities by advancing or outright giving the team money to make critical purchases.
Chaperoning trips is often handled as a separate issue by the school administration, because teachers have to get substitutes to cover for them in classes that are missed. If a critical AP test is coming up, then someone else may have to be found to chaparone. There is also usually a chaperone/student ratio, e.g., 10/1, that requires the recruitment of additional teachers, so teacher contracts or districts have set fees they pay to recruit the necessary chaperones. Supervision and chaperoning of the predominately underage students is a legal responsibility, and is a firing offense if not done with due diligence.
I know some administrations struggle to get teacher's, who have to volunteer to begin with, to dedicate the enormous amount of time that a FIRST team requires. A teacher may only reluctantly sign up to do the minimum necessary, and they may only end up a temporary place-holder while the team looks for a fresh, young, idealistic replacement. It's much easier to lead the school fly fishing club, and the time and expense commitment is much more reasonable for a person who probably has their own life.