You have the following goals:
(I) Every student will transcribe at least one lecture.
(II) The transcribed lecture notes will be of a sufficient quality to be useful to the other students, and ideally to be posted somewhere for others to access.
(III) You don't want to spend an unreasonable time writing or correcting the notes yourself.
In my opinion these are three worthy goals and any one of them is attainable, but as a set they are probably incompatible with each other. So you may want to choose (not necessarily once and for all, but depending on the course) which of these worthy goals is the worthiest and prepare to sacrifice one or both of the others. In more detail:
If you prioritize (I), then the emphasis is placed on creating a learning experience for the students. If you do this you will find that the variation in the quality of the note-taking will be amazing. I have seen this in students' written notes over the years. Yesterday a student left his notebook in my office. I flipped through it today to try to figure out who it was. It is one step above chicken scratch: for instance, he is writing on lined paper but his mathematical expressions and equations only stay within the lines about half the time: he can have a "written slope" which is only slightly larger than -1. (This helped me identify the student, who does not have his name written down anywhere. I noticed his characteristic negative slope when he was writing on the board in my office yesterday.) On the other hand, in my graduate course I have glanced at several students' handwritten notes at the end of the lecture and marvelled at how much better they look than my own handwritten pre-lecture notes and what is on the blackboard at the end of the leecture. At one point years ago I gave in a borderline undergrad/graduate level class what I thought was one of my best lectures of all time and at the end realized that I hadn't prepared any lecture notes and sadly reflected on the fact that it would be lost forever. Then I remembered that I had one student in the course who took great lecture notes, and I asked her for her lecture notes for this purpose. I think that what she gave me was an improvement on my delivered lecture, and using her notes I got my favorite part of what amounts to a textbook on number theory: you can see the lecture notes here. (Yes, I touched it up a bit afterwards, but not by that much.) You can also see that I thanked this student at the end of my notes.
All of the students mentioned above are "A students". I personally do feel that the gender difference between the students is playing a role here. This is not a scientific observation and I certainly mean no offense to anyone, but in my experience the difference in note-taking care between male and female students is, on average, dramatic.
Anyway, if you prioritize (I) and (III) you're going to get such uneven samples as to make it a pretty jarring experience for anyone to try to follow the entire course by reading the notes. Having the notes be texed adds other levels and brings other skills into play. Some people will naturally go back and spend time prettying up their tex files; others won't. (I started out as the latter and very slowly over the course of thousands of pages of notes am heading towards the former. But for someone who has thousands of pages of texed mathematics on his webpage, my texing skills are distressingly middling.)
Thus if you prioritize (II), then I think you need to seek out the students who will do the best job. This works against (I), and you need to look for extra compensation. I agree with the other answers that providing good latexed notes is something that may be worth paying for.
If you really prioritize (II) then at a certain point you will decide to type the lecture notes yourself. This is what I do in most classes I teach. At this point I have a bit of a reputation as a guy who has lots of typed lecture notes. When I begin a graduate course now, I often tell students explicitly that I will not be typing lecture notes as the course progresses, because that is very time-consuming. But then, at some point in the semester I usually break down and reveal to the students where my heretofore secret lecture notes can be found. I actually just did that yesterday in my graduate class this semester....with less than one week of class left.
Typing your own notes totally destroys (III): to get something that looks good you have to work significantly harder during the semester than you otherwise would need to do in order to deliver exactly the same lecture and you need to put more time into it after the semester is over. On the other hand, I have found this practice to be extremely valuable to both my learning and my career: I have gotten a lot of wonderful professional interactions out of it. Thus I would say that the real drawback is that it defeats (I).
To summarize a very long answer: I think you need to decide whether the point of this is the process or the end result. You either provide a learning experience for all the students or you provide decent-to-high quality lecture notes, or you provide both but don't get it done until after the end of the course. I don't see how to do everything at once.