I already answered nearly the same question. (What, is that the school assignment, the same one for a group of students?!)
Basically, there are some variants: 1) custom Windows Service on each student's computer, 2) simple HTTP server on each student's computer, 3) some available HTTP server with Web application on each student's computer. In cases (2) and (3) the teacher's UI could be just an available Web browser!
For more detail, please see my answers here:
Remote Screen Capture in WPF[
^],
Multiple Remote Screen Viewer in wcf / wpf and c#[
^].
Note: It does not have to be WPF.
—SA