How can a student make private contact with the teacher/s?
If there is only one teacher, the student can send mail or a Moodle message to the teacher.
If there is more than one teacher, one could set up a shared email account outside of Moodle, to which all teachers have access, and instruct students to use it.
The drawbacks are that Moodle messages are outside of the context of the course and email is outside of Moodle altogether.
A better solution is to open a Personal Forum (ForumNG) and instruct students to use it.
(Consider setting option Email subscription to Force everyone to be subscribed or Everyone is initially subscribed so that students will be sent your by email.)
Another solution, for private discussion on a particular Assignment, is to use the assignment's Submission comments.
This has the serious drawback that a student has no way to alert the graders when he posts a new comment, and graders have no easy way of finding comments that they have not yet responded to.
Another possible solution is to create a group for each student (with Auto-create groups) and use the regular Forum with Group mode set to Separate groups and Grouping to the associated grouping.
The only drawback is the inevitable plethora of groups.