Well... by the rules, sometimes yes, and sometimes no.
Ok, let's look at a scenario where someone actually needs both hands. A boxer. This is the default ruling that opens up the whole "left hand AND right hand as an advantage" line of thought. It is pretty natural to swing one hand then the other. It's like walking or breathing. My house rule is that left and right punches or kicks have no bearing on the game. There is no penalty if someone kicks left and punches right, even in the same round. The motion lends itself to the practice.
What about a sword fighter? Well, to be honest, fighting men probably bust their wrists a lot, so it would not be implausible for them to train as if ambidextrous. Archers train heavily and off hand shooting would be something they would explore as a gimmick or challenge. Wizards and clerics casting spells probably causes them to do the same.
So, most of your characters should be ambidextrous... to an extent. It might be plausible for a wizard to write with either hand because all of their magic is focused on hand movements, but a fighter isn't going be especially noted for writing, period. Fighting, the focus of most games, should not incur a penalty for using the off hand. Hockey players, baseball players and other athletes typically have the ability to use their off hand, sometimes even with the wrong type of tool because they are professionals. And all RPG characters are professionals, so the logic neatly follows.
However, using your off hand for other, uncommon tasks should incur a -2 penalty. Say, painting a picture with the off hand, especially when your character isn't a professional artist.
What the players are really asking for is a second attack because the character has two hands. The rules are punishing, a significant penalty because this is a gimmicky move. And unfortunately, these rules get abused by both players and DMs. When should it apply?
When the player is demanding a cheap and easy advantage for goofy reasons the negatives should apply.
History is full of cases where people walk into combat with two weapons. The player abuses this fact. A soldier with a sword and dagger will likely use one item to parry and one item to attack, which one depends on the circumstances. And it's fluid, they switch back and forth. For example, a swordsman may make an obvious "attack" with the sword only to swing the dagger in because the sword was blocked. That is one attack, the sword wasn't really swung with the intent to hit, merely to tie up the opponent's attention and weapon. The DM should be aware that this was relatively common, so she or he shouldn't want to invoke this penalty at this time.
How do you simulate this?
Option one: Average weapon damage. If a character has two daggers, then they do (1d4+1d4)/2. If they have a long sword and dagger, they do (1d4+1d8)/2. At first glance this looks odd. How can a dagger do 5 points of damage? We have two weapons and there is a small chance that both land hits, but one of them was far less effective than the other because it wasn't swung with intent. To this end, if a 20 is rolled, both weapons have hit but instead of double damage, the player merely rolls damage once for each weapon and adds them together. It could be great or could be poor.
Option two: Have the player declare which is the parry weapon, forcing them to roll for damage with the other weapon. In this case, the player has the option of either weapon and is technically declaring which hand they are using without saying so much. Damage rolls for a 20 are back to the normal double. Technically, they are handling two weapons, but don't have a chance with one of them.
If they player insists they can swing both weapons at the same time, this is when you start piling on the off hand penalties. They aren't entitled to two attacks because they aren't skilled enough. So add those penalties up.
Now, what happens if the character IS entitled to two or more attacks? Nothing in the rules says that the player can't swing one mace 3 times in a round, nor does it say they can't kick someone as an attack in lieu of swinging the mace. Let them do it. Options one and two can be combined with this, if the character is so armed.
How do you handle two weapon attacks in your campaign? Let me know in the comments.