In VATUSA's Training Resource Center, under VFR Tower, it states
This is incorrect unless all of my instructors were wrong. The Tower controller should keep VFR aircraft within the Class B/C surface area (usually 5 mile radius) of the Class B/C airport instead of handing them off as you would IFR. If the pilot was VFR doing closed traffic, the Tower controller would not hand that pilot off and would maintain control of that aircraft, such as advising the pilot to fly left traffic, report mid-field downwind/base/final or whatever. Otherwise there is a lot of unnecessary handoffs back and forth between TWR and APP/DEP or CTR. I know I have had 4 or 5 doing closed pattern work in the ORD Class Bravo airspace before and if I would have handed each one off to APP or CTR and then have gotten them right back, there wouldn't be much of a chance to control anyone else do to the constant handoff/acceptance, etc.
This quote can be found here.
Your instructors may have been misinformed. This is from the 7110.65 regarding communications transfer for departing aircraft:
"3-9-3. DEPARTURE CONTROL INSTRUCTIONS
Inform departing IFR, SVFR, VFR aircraft receiving radar service, and TRSA VFR aircraft of the
following:
a. Before takeoff.
1. Issue the appropriate departure control frequency and beacon code. The departure control frequency may be omitted if a SID has been or will be assigned and the departure control frequency is published on the SID.
PHRASEOLOGY
DEPARTURE FREQUENCY (frequency), SQUAWK (code).
2. Inform all departing IFR military turboprop/turbojet aircraft (except transport and cargo types) to change to departure control frequency. If the local controller has departure frequency override, transmit urgent instructions on this frequency. If the override capability does not exist, transmit urgent instructions on the emergency frequency.
PHRASEOLOGY
CHANGE TO DEPARTURE.
3. USAF. USAF control towers are authorized to inform all departing IFR military transport/cargo type aircraft operating in formation flight to change to departure control frequency before takeoff.
b. After takeoff.
1. When the aircraft is about 1/2 mile beyond the runway end, instruct civil aircraft, and military transport, and cargo types to contact departure control, provided further communication with you is not required.2. Do not request departing military turboprop/turbojet aircraft (except transport and cargo types) to make radio frequency or radar beacon changes before the aircraft reaches 2,500 feet above the surface."
The only exception with Class C is if the tower's control area is the same as the lower portion of the Class C airspace and the pilot will exit the tower surface area without breaking the upper portion and does not wish to have flight following. Then you, as a tower controller, can hold on to the aircraft until they are out of your airspace. However, in many cases, tower's airspace is smaller (Towers usually own 3-4 miles of actual airspace in most cases) than the inner portion of the Class C and therefore, frequency change is required to departure (as radio communications is one of the requirements for operations inside Class C airspace). Same exception applies to Class B airspace where the aircraft will exit Tower's airspace and be outside/under the Class B so long as they do not want flight following and remain VFR.
In the end, yes, communications transfer is required in any case where ATC communications are required and/or radar services are to be rendered around 1/2 mile beyond the runway end except as prescribed above. Also, you can keep them on your frequency if further communication with you is required (IE, you have a departure that may require separation from aircraft in your tower pattern or transitioning through your airspace then you may keep them a little longer). If communications is not required, 7110.65 requires communication transfer for all aircraft receiving radar services.
Additionally, as far as the traffic pattern, aircraft in the tower pattern at not departing out of the Class B, so they are not bound to talk to departure unless they leave tower airspace. Take it as simple as this:
If the aircraft is leaving tower's airspace and it's not a safety of flight issue, then they are to be handed off to departure around 1/2 mile from runway end. If they are staying in your airspace, they stay with you until a) communications transfer required [aka, going to an uncontrolled or controlled airfield within or immediately bordering your airspace], they land, or c) they change their mind and request to leave your airspace.
For me, if they are not doing a straight out departure and are doing a downwind departure or base leg departure, then I will keep them under my control until they are clear of my pattern for safety reasons unless I see no one else in the area because they are in my tower pattern until they have departed it VFR. This is all interpretation tho.