Can you cite other master's theses when writing your own?

In general, there is no such widespread requirement or expectation anywhere in academia. (Perhaps a particularly unusual advisor might have such an expectation, but even then, I doubt it.)

If you note the citations of research papers, a decent number of them will cite master's theses—some of my papers certainly do! And if I can submit a paper to a major journal that cites a master's thesis, why wouldn't another student's master's thesis?

Yes, you can cite another master's thesis. You also should, if it is relevant to the work. And no, don't just pillage that other thesis for sources and ignore it. That would negate the benefit of access to that thesis, which you should make available to future readers of your own thesis.

Above all, don't think like a student. Think like a writer, which is what you are. Your work, and those of other writers, should be treated with due consideration no matter the pedigree of the writers. Stop thinking of academia like some sort of sacred place where "students" are lesser beings. There are no gods here; only flawed men and women with nary a grasp of purpose.

Firstly, I would ask your supervisor/advisor for advice.

But, generally speaking this should be fine, as Masters thesis are examples of completed and verified research (in that they have either undergone defense or are peer reviewed - as was my case).

One thing you could do though, is to use the thesis almost like a well-referenced Wikipedia page - in that you find the main referenced points that are relevant to your research and seek and peruse the paper that was used - chances are, you may find more information to assist in your specific research.