Here are my answers, from the perspective of an undergraduate majoring in computer science and engineering. The engineering college provides free tutoring 6 days a week from 6pm to 9pm.
Are the tutors other students or professionals?
The tutors are all other students.
How are the tutors selected?
The tutors apply and are interviewed. The interview consists of one prepared problem and one unprepared problem, both taken from areas they claim to be able to tutor. They then have to mock tutor their interviewer.
Are tutors or sessions supervised formally in any way?
No. Students come in and ask for help in a subject, and tutors assist them. If a tutor has nothing to do, they usually do their own homework.
Are tutors paid? If so in what way? (money, academic credit, ...)
Yes. Tutors make somewhere around \$10 an hour to $15 an hour, depending on subjects, expertise, etc.
What are the scheduling issues? What works and what does not?
There are none that I am aware of.
Are the tutors and tutees matched with each other in any specific way?
No -- it's entirely informal. You do not need to sign up for an appointment or anything, you just walk in.
Is tutoring one-one or one-many, generally?
It depends on the demand and subject area. Prior to exams or important homework assignments it will usually be one-many. Less common subjects attract fewer students.
Does the student need a "ticket" to join, say a recommendation from a prof?
No -- it's open to anyone.
Can you get any measure of success?
I'm not sure what is meant by this. University analytics seems to indicate that students who go to tutoring do significantly better than students who do not. Otherwise, attendance is logged, and is usually fairly high -- showing that students clearly think it helps.
How do the tutors benefit other than formal pay?
On off weeks, tutors often get plenty of time to work on their own homework while getting paid. Additionally, if they are part of the honors program, it can count for credit in that regard, and tutoring is a great way to become a TA later (and many of the tutors are also TAs).
I think the way the engineering college has set up this system is an efficient use of resources. It's lamentable that our regular CS program does not have anything of the sort. The CS department maintains a list of approved tutors (anyone can apply online, fill out a form, and be approved as long as they meet a sufficient course grade for the courses they want to tutor). There is no formal training whatsoever and close to no oversight from the university itself. This results in incredibly variable tutoring experiences. Additionally, the system itself is poorly known, and I doubt most students would be willing to pay a private tutor 15-30 dollars an hour to quite possibly get no help other than a brief explanation of the answer to their assignment. Of course, this is speculation on my part, but I think if the computer science department were willing to budget some money to set up a system like the one I've described, it would be a great investment. It would ease the load on TAs and professors (as an aside, I recall prior to one homework deadline in my data structures class, my professor and TA for that class had 30-40 people lined up for an office with four chairs), ostensibly reduce cheating, and improve understanding.
Of course, that's just my opinion -- but I have no reason to believe the former model would not work for common computer science courses as well.