In a nutshell: no, don't require note taking, don't structure the course to really need it.
I think this is very much something where different methods by professor or student can work, so not the canned input-output Q&A format, but more a salon-like different ideas type of question (as many here are).
That said, my $.02 would be to use a book and drill program with a lot of scaffolding. NOT the "don't need a text, just follow me" idea of the romantic lecturing college professor--rather more of an athletic coach or professional trainer instead. Lectures should introduce the topic with a a question and a demonstration and then give some examples. Rest of the time is student questions, calling on students, recitation, drill, tests, etc.
I would not bother with publishing notes, especially if you are using chalk, which seems the most human way to communicate. (Not the acetate and OHP, not the projected laptop...at least you write on the fly, versus PPT, though!)
Would encourage students to take notes as they see fit, but again try to use a book that is good at explanation and examples, for those who gaff it off or miss something. And follow it. It is better to do something good, perfectly, than to do something perfect, poorly. And you have to realize there are other humans in the loop (the students) and they are the bottleneck. Which is my constant reminder to the rigor-luvvr curriculum questioners. Sometimes a shorter text, that uses more accessible language...vice impressing professor buying committees...is the way to go, but that is a tanget.
As a student, I had a system (got from a friend) and has helped me in every single class: Have a separate spiral (or bound) notebook for each class. Drill homework, or example trials (from the text) are "normal pages". Lecture notes are separate pages and have inked stars at the top. Proceed left to right and intersperse as the class progresses. Stars allow checking notes. From the back, use this for questions (with spaces for answers). Forces a feedback loop to ask instructor (sometimes figured out on own, later, though). A separate file folder (or a section of a 3 hole binder, but then you need a punch and reinforcements) is used to store handouts, tests, etc.
Best practice is to pre-study the material. In this case, lecture is really a dream (a review) and notes are easy and less of a transcription. Would encourage your students to do this, but realistically few will because of human nature. But even there, I would emphasize to them that you have a good book, that the tests WILL be from material in the book and that the lectures are so they have one more way of learning the material, not as something they must hang on since material outside the scaffolded text is covered. [IF you have some cute method like tabular integration, fine...add it, but not as a tested requirement, more as an "easier way to do things".]
P.s. I'm not a teacher. I just took STEM math classes (and not majors courses) and watched Stand and Deliver. And lived it. (Sort of.)