Grammatically it doesn’t matter. However, logically one should say “on one hand … on THE other hand.” That’s because when you pick the first hand you have two to choose from. So “the” one isn’t logical. It’s one out of two. However, once the first has been chosen, there is only one other one, so it is THE other hand. It’s mathematics, not English.

Either is fine:)

The full complete contrasting sentence requires, or at least used to require both. On the one hand, I would like to eat at an expensive restaurant, on the other hand I cannot afford it. You state one situation or fact, (on the one hand) then you state the other contrasting situation/fact/situation/opinion (on the OTHER hand). Over time the use of both as one long sentence/grammatical structure has declined. So that now many people use it with only one of the two. And often with no contrasting ideas/opinions/facts/situations. I am not sure if there is any difference between British usage and American usage. But I doubt it.

The correct phrase is "on the one hand". The phrase is typically used to introduce two contrasting ideas or options. For example, "On the one hand, I would like to go to the party, but on the other hand, I have to study for my exams."

Both are acceptable, but "on the one hand" looks more correct.