On the one hand on the other

No.

In Chinese, it says 一方面...另一方面

so you could say, "On one hand, i really want to participate in the singing contest; one the other hand, i am really shy to sing in front of people"
一方面, 我很希望參加歌唱比賽, 另一方面, 我不敢在人前獻唱

or "On one hand, i am so full; one the other hand, i want to try the blueberry ice cream"
一方面, 我很飽; 另一方面, 我想嘗嘗那個藍梅雪榚

So, they are usually used in a interrelated content.

On the one hand I would like to have a job which pays more, but on the other hand I enjoy the work I am doing at the moment.

On the one hand cars are very useful, but on the other hand cars cause a lot of pollution.

On the one hand I admire your ability, but on the other hand I distrust your judgment.

On the one hand, we think the price is fair, but on the other hand we really can't afford to spend that much money.


Comment:
~ used when you are comparing two different facts or two opposite ways of thinking about a situation.

2012-11-12 09:13:38 補充：
參考資料: Cambridge English Dictionary

on the one hand on the other(hand) is used to introduce different points of view, ideas, etc. especially when they are opposites (used) in the same subject (on your hands!).
eg:-On the one hand they would love to enter the University,but on the other (hand),they don't want to work hard.
eg:-On the one hand they would love to have children,but on the other(hand),they don't have any money!