Well, it has no bearing in terms of number crunching, so I know a lot of players that don't worry about it. Any weapon held in one hand takes no penalty if you are proficient, whether the left or right. Two weapon fighting incurs penalties, depending on the feats or whatever you have (assuming using 3e or higher).
That said, I personally assign my character a hand on whim at the start of a campaign. I take a voluntary -2 penalty if he uses a weapon in his off hand. Occassionally I alter this, based on setting and such (Celtic warriors were known for fighting with either hand equally well in mythology). But I never get quite that into the "rules" aspect of it.
When you create your D&D characters how do you determine if your character is left or right handed? I don't know if this is taken care of in later editions but there was nothing in the first edition so we used a d6 and a d12. If the d12 was higher then you were right handed, obviously if the d6 was higher then you were left handed however if both came out the same then you were deemed to be either handed. if both die rolled a 6 then you were ambidextrous with no penalties for using two weapons.