Well, I'm far from an expert on this topic, but I can make suggestions based on what has worked for us.
First, the question of money.
The best way to do this is to have the instructor buy the mats out of pocket (about 125 for a lot of 20, including shipping) and then after class one week, have everyone roll and tie the mats (I use size 64 rubber bands, a trick I learned in Toyama Ryu). Then take them home and soak them, and bring in two or three mats per person on cutting day (once or twice a month). Charge each person what the mat is worth with a small overhead for rubber bands. This will be about 7 dollars per person, per mat. Since each mat yields about five cuts, not including the fragments, that's a little more than a dollar a cut, and I think that's quite economical.
If you have no place to roll (if say, you practice outside), the instructor can roll all the mats himself (or herself) and charge a bit more per mat, so that the reward for the work is some free mats every cutting session (if you charge 10 a mat, you can have 8 free mats out of 20 to make up for the hard work of getting them ready). Or one of the students could do this instead.
Or some combination of the two above. Either way, each student pays a little bit every now and then to cut.
Now the question of how and when to do it.
The first thing you need is an instructor that knows how to institute and run a cutting curriculum. For that, he/she has to understand the purpose of cutting and how best to approach it. For example, if you don't understand why many JSA schools cut from a standing position (hint: it's a pedagogical thing), you probably don't know how to get the most out of a cutting program. The best solution for this is to find a local JSA school that cuts and enroll. Failing that, find someone to help you, over the phone (too much to explain in text), or to invite over for a seminar, or whatever works best for you. Keep in mind that if you ask me, you will be getting a photocopy of a photocopy. I am a low budget Japanese parrot.
Better to go to the source and find a local JSA instructor willing to help you (shouldn't be hard, if he is approached with proper respect).
I am not at all suggesting we adopt a Japanese program wholesale...we practice a different art and there will be differences. But the pedagogical methods employed by the better JSA schools for cutting are priceless and simply cannot be arrived at by trial and error in anything less than decades.
Once you have an instructor who knows how to run a cutting program, your next problem is swords. Many European swords don't cut flesh very well (and by extension mats), and when someone is learning how to cut, you don't want a sword to get in their way. The Albion Crecy is one of the best cutters on the planet (out of the European swords, anyway) and is not terribly expensive (about 600 bucks on the used sword market). It would be a worthy group purchase. Also, if you can find older Angus Trim longswords, or the newer CF/AT line, some of those work very well too.
Finally, you have the question of when. I start people right away, though I moderate how much they do. Start slow and simple.
I hope this helps. If you have any other questions, feel free to ask.
Here is a link to a cutting seminar handout from our website: http://newyorklongsword.com/articles/cm.pdf