First, as to the overall question about posting information about grade distributions on assignments: I do not do it. More below.
Can one do better? Information about quartiles maybe?
One problem that you may be trying to solve to give better information to students on how to map a score to a true course grade. In other words, you may be trying to avoid the anxiety of a "hidden" curve grading system from the perspective of the students. Taken to an extreme, you are wanting to avoid the case where your students don't really know where they stand against your grading metrics until you post their final course grades.
Is this is where you are trying to do better? If so, perhaps you should just post the equation that says directly:
Assignment Score --> Assignment Grade
When this is still unknown for some reason, then at least post a statement to the effect:
Assignment Score --> AT LEAST this Grade
This is NOT the grade distribution of the students in the class at that point on that assignment/exam. This is how you will map any given score to a likely future grade going forward.
Perhaps better still, fix the curve on the assignment grades before you report/return the score on the assignment. Add offsets or rescale. For example, rather than returning an exam with a 50% "hidden curve" grade and having to report that 50% --> B, rescale the exam scores so that 50% becomes 85% up front on the exam.
In short, when you are trying to avoid causing undue anxiety in students because the truth in your curve grading system is "hidden" to them, then stop using an after-the-fact curve grading and start reporting a "truth in grading" score.
Another improvement that you hint that you are trying to make is in your approach to motivate individual students to do better. Let's presume that you are solving this problem separately from the above problem. In other words, let's presume that, when a student gets a score of 85% on you exam, that student knows unambiguously that they have a B on the assignment. Correspondingly, a student who gets a score of 67% on your exam knows unambiguously that they have a C- on that exam. Let's presume that both students know that these grades are not subject to any future "hidden" curve grading (leaving aside case-by-case benefit of the doubt situations for borderline scores).
In this case, I must say that, I have never heard nor believed in a philosophy that a posting a grade distribution does any good or even more good than harm to motivate an individual student to do better compared with other options. In some cases, e.g. small class sizes, I even fear that posting a grade distribution, even in part, gives enough information for students to reverse-engineer grades for other students in the course. Being in the US under the guidelines of FERPA, I tend then to just avoid any possible downside risks, in this case that Student Joe will be able to figure out the grade for Student Sally. I do not post the grade distribution at all. You asked about practices as a function of class sizes. Perhaps this counter thought will give you also some insights to the potential harm.
What are your practices?
My first practice is to follow a "truth in grading" philosophy. I post the grade metrics before the course. I state where the cutoffs are for the various grade levels. Finally, I state that all problems on exams will be graded for completeness using the given grading scale.
What happens in practice is interesting. At times, I give somewhat easy exams looking back. I leave the scores fall as they do. On the next assignment or exam, I may push the boundary a bit higher for performance. At times, I give rather difficult exams, in some cases perhaps to a degree beyond what was appropriate. At that point, before I return the exam, I rescale all exam scores for that exam. In class, I state that I rescaled the exam because, taken on a wider perspective, it was more difficult than appropriate. I do this also on individual problems on exams, dropping or adding points back because I realized that the specific problem was (in hindsight) too difficult.
The net result (after many years of practice) is an ability to grade assignments and exams with final scores that are not subject to a need for a "hidden" curve to be applied later. This practice still allows for benefit of the doubt considerations for borderline cases. However, again by truth in grading, I am adamant to tell students that they will never get a lower score than what they have directly calculated, although they may (in borderline cases) have shown initiatives to obtain a higher score.
My second practice is a message to students that they are not in competition with each other or anyone else but themselves in my course. By specific example, I state that an 85% is a mid-B grade and, should this score be the highest returned grade on a course exam, then that is what the highest grade on that exam will be. I also purposely turn off the ability for students to see the overall course grade distributions in our learning management system (Canvas).
My final practice is a message to students that individual initiatives matter. Along this line, I am apt in junior or lower level classes to give bonus points for office visits.
I am interested in keeping the students motivation high. Are there any (research backed-up) studies about that question?
I too would be curious to hear whether giving out grade distributions motivates students or not. I would especially be curious to hear in the case that "hidden" curve grading is removed as a factor in such studies.