Either is fine.
I'm not aware of any regional differences in usage.
Edit two years later
I probably didn't think very carefully about this answer at the time, but given the subsequent comment and other answers I am satisfied that it is still correct.
My answer, in case of doubt, refers only to the use of the expression when used to compare to contrasting things (as the OP requested).
Either of these two options is completely idiomatic and acceptable to native English speakers, even if they don't realise it.
One one hand ..., on the other hand.
On the one hand ..., on the other hand.
Omitting 'the' before 'other hand' is not idiomatic, at least in American and British English. In fact it makes logical sense that 'the' is required in the second part of the comparison. The 'one hand' could be any generic hand (no article) or a specific one (definite article). 'The other' refers to the 'matching hand' of the first one. There can only ever be one, and it is being referred to explicitly as the partner of the first, therefore it needs a definite article.
While I'm still not aware of regional preferences towards one style or the other, it seems likely from comments that they do exist. I don't think it is an North America/Britain distinction, however, and may be highly regional. For instance, I know that in Ontario 'the one' is not only preferred, but sometimes used in a way I find odd to refer to a member of a group. Eg "The one student is very badly behaved" where I would expect "One student is very badly behaved". In this context "on the one hand" would also be likely to be preferred.