Many PLAYERS attempt to gain an advantage by swing two weapons at once. The various sets of rules accommodate this in one of two ways:
1) the character is simply swinging two weapons very fast, like a skill-less boxer.
2) the character is high level and receives extra attacks.
The first has a significant penalty, while the second does not.
My house rules handle these events slightly differently. Unarmed characters, who are not monks, can punch or kick twice per round. They receive no penalty unless they are making a last attack after losing a morale check. Failing a morale check removes any strength bonuses as the character is panicked. They are kicking and punching to get away, not to do damage.
By the way, punches do 1d2 points of damage per hand and kicks do 1d3.
Having established that it is natural to attack with both feet or hands and that panicking is bad, we move to the next scenario: Weapons. If a player has a weapon and a free hand, they can attack with the weapon and punch or kick at the same time. There is no penalty, as being unshielded is a penalty enough.
Characters may use a shield as a weapon, but they operate paradoxically. Bucklers do 1d3 points of damage plus strength bonus, while larger shields are relatively ineffective as weapons and do not do damage. A large shield, although it does no damage, it can disrupt a spell casters ability to cast and can foil a missile attack against another character. To hit someone in this fashion with a shield, one must normally be able to use a shield. Fighters, clerics, assassins, etc.
Now let's add in that second weapon. A character with a long sword and a dagger, 1d8 damage and 1d4 respectively, can use the tempo of combat to strike or threaten with both weapons each round. Only one attack roll is made. The effect is that the damage is shifted to a point between the two weapons: 1d6. Maybe they hit lightly with both weapons or perhaps they used one to force an opponent into dropping their guard for a single attack with the opposite weapon. Which one happened is not important, we are merely empowering players to act out realistic scenarios as they see their characters behaving. If a character is using two equal weapons, say two short swords or a mace and short sword which do 1d6 each, there is no change to the damage. It is simply a 1d6 roll.
In the AD&D rule set, there are significant penalties to swing both weapons at once. These rules should remain in effect when a character is unskilled, panicked or otherwise unwilling to be subtle in their attacks. Given the choice of using combat tempo or taking a big penalty, when would a character desire the penalty? Attacking massive creatures that can't strike back, attacking from behind, attacking creatures where a little damage is worse than no damage.
The last scenario, when a fighter gains multiple attacks can be handled either by the combat tempo rule which reduces damage, OR they pick one weapon to swing OR they can do the crazy "two at once" swing with a penalty. The player should choose based on the needs of the situation.
Did you know I have a little book called Zero to Hero: Uncommon Commoners, over at DrivethruRPG? It contains all kinds of rules you can use in your campaign. Give it a try, it's pay what you want.