Not necessarily. You can omit it if you want. For example: Making mistakes is not always a good thing. On the other hand, you learn a lot from them; that's how you make progress.
I remember that a teacher of mine forced us to do it. We were obliged to have said "on the one hand" before using "on the other hand". Overkill? I don't know. If you are writing a very formal text perhaps you should use it. There is no harm in doing it.
Neither a teacher nor a native speaker. ;-)
PS My teacher was English native speaker. From the States to be precise.
Browsing around I have found one of the post sent by bhaisahab with the title Stranger than Fiction - 09-Feb.-2010. He uses "on the other hand" without having used previously "on the one hand". In case you may be interested......;-)