Is it appropriate for professors to ask students to type notes for their classes (with or without paying students) in United States?
It would be unethical for a professor to publish something written by students under the professor’s name without express consent (which is to be given free of any coercive pressure) and coauthorship for the student. (See this recent discussion on a related type of abuse.)
It would also be self defeating and something that no competent professor I’ve met would ever wish to do. Students do not generally write notes at a level that comes close to the level most professors can write on their own. Publishing someone else’s subpar notes under your own name is a sure way to hurt your own reputation even if you do technically get some publication credit.
Setting aside the publication issue, it’s fine to have note-taking as an official course assignment if there is an educational rationale, which there often is. For example in mathematics graduate programs we try to get students to work on improving their mathematical writing skills, which are extremely important for success in research, and some writing assignments of this type are not uncommon.
If note-taking is not an official course assignment but the professor simply asks students to do it on a voluntary basis, the request should be made in a way that makes clear the students should feel free to say no. To put pressure on the students to do things they don’t want to do and that are not an explicit course assignment is unethical, regardless of whether the professor publishes the notes or not, and regardless of whether some payment is exchanged.