FCC 47 CFR § 97.5(b)2 lays out the structure for a club station license. In practical terms, this gives amateur radio clubs a memorable callsign of note they can rally around, to build branding, etc. The club station license does not confer operating privileges, but the trustee must hold an amateur radio license.
The trustee will receive FCC official mail, and ARRL LoTW (Logbook of the World) will only sign up clubs through the trustee. Thus clubs should ensure the trustee is someone who actually checks their physical mail, that is on campus at least several times a month on average. The trustee does not have to be be a school employee. However, some club constitution or bylaws require the trustee to be a school employee.
Control Operator vs. Club Trustee: §97.103(b) notes that by default, the station licensee (here, the school club trustee) is the control operator. Even if another person is the control operator, §97.103(a) holds the trustee and control operator equally responsible. Intuitively, §97.105 states the control operator is responsible for “immediate proper operation of the station, regardless of the type of control.”
Student operating privileges: provided the control operator is in control, anyone operating (including non-licensed persons) may use the control operator’s license privileges. Ideally, the club control operator will have an Extra class ham radio license so that operators get to use maximum privileges. §97.115(b) notes that for third-party communications (someone besides control operator working the radio), the “control operator is present at the control point and is continuously monitoring and supervising the third party’s participation”. Many clubs take §97.115(b) to mean the control operator is physically on site, indeed in the radio room itself.
Reference: FCC § 97.5