Kingdom Animalia, 'cause boobies, we got em.
The genesis of the idea was for an official website for clients to find tutors. Furthermore, if the tutor set it up to work this way, they could auction off their sessions to the highest bidder. This would work well for a tutor who worked out of a public area, like a library. It could also just work as a place for a tutor to post their schedule and their subjects so that clients can find someone who works for them. Another way to connect clients and tutors would be that if a client couldn't find someone to suit their needs and schedule, they could post a bounty for tutors to find. The last way I can imagine would be to hold a webcam tutoring session.
The site would have a rating system, so that people would know if a tutor was bad. Furthermore, the tutors would also (secretly) rate the students to know if the student was trouble or not. They could also post comments about the student's problem areas and progress that would be visible by the other tutors.
As with anything like this, there are issues. Warnings would have to be posted and reminded about regarding safety, such as meeting in public places or at your own residence and having adult supervision.
There is the issue of monetization. Ads are a definite way to monetize. Centralized tutoring services (such as Sylvan) could advertise on the site. Textbook/ebook sales are another definite way to generate revenue. The other way would be to have payment be handled by the website and have a commission for each session plus maybe a monthly fee for the tutors (only if they book a session that month, to remove the barrier to entry).
The risk with handling the money is that now the site becomes responsible for it. If the tutor doesn't show up, they should be paid. If the student doesn't show up (without canceling ahead of time) then the tutor should still be paid. A system would have to be put into place to verify that the tutoring session happened/didn't happen and why. My best idea would be to have an email be sent to the tutor that, when opened, would have a link to the site. Once there, the student or parent would have to enter a special code/password to verify their attendance. I still don't have the logistics of it worked out so that it's foolproof. Another is to have some sort of geolocation tied to IP addresses or maybe a smartphone app tied to the GPS to verify locations. It's rough idea.