It depends.
If someone comes to you and tells you about it they obviously trust you, and for you to tell someone would be breaking their trust. But if you're truly concerned I understand it's hard to handle alone, as their sole "mentor"--I've been in both positions. In that case you can encourage them to speak to someone or offer to tell someone if it's easier for them not to do it themselves. To just tell behind their back is betrayal.
That said, when my parents found out about my self harm, I didn't want ANYONE to know. One of my best friends accidently caught sight of my cuts, and figured out what they were. We talked about it and I made her promise not to tell but she did anyways. She told a teacher and the two of them confronted me and forced me to tell my parents. At the time I felt extremely betrayed by her but I realized today if not for her, who knows when, if ever, I would have gotten help for my depression.
So, it depends on the situation.
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