You do need a second weapon to employ two-weapon fighting, but the attacks do not need to be made with alternating hands. The second attack must be made with the off-hand, but the first attack can be made with either hand, subject to appropriate penalties. When you attack twice with your off hand rather than employing your primary hand, be aware that you incur the two weapon fighting off-hand penalty on attack rolls on only the second attack, yet incur the off-hand penalty to damage on both attacks. This is probably because the rules very much don't expect you to do that. You do take a -4 penalty to hit to the first attack for using your off hand, like normal, though.
Since you must wield two weapons (or have something special that bypasses that requirement, like a double weapon) to make use of two-weapon fighting, and the off-hand weapon must be the one to make the extra attack, passing a weapon between hands isn't a very viable option. It does work for thrown weapons that can return to a different hand, though, as a special case, though you need to use at least two attacks before you can benefit from Two-Weapon Fighting on any given round in that case.
Basically, if you play strictly by the rules as written, you can do your melee two-weapon fighting at -4/-10 without the feat or light weapons, or -4/-8 with just a light weapon, or -4/-4 with just the feat, or -4/-2 with the feat and a light weapon, with the bonus of using your off-hand weapon twice rather than two separate weapons once each.
(Note that the action cost of passing weapons is weird and controversial. The rules don't actually say anything about it, but Rules Comprendium lists it as a move action, while an earlier Rules of the Game article says switching from one to two hands and vice versa are free actions, which would make passing possible as a free action as well. Pathfinder codifies the Rules of the Game version into its actual rules, while the Rules Comprendium, while not an authoritative source, is probably more authoritative than a Rules of the Game article for 3.5, in general. This doesn't matter because passing weapons doesn't work unless you are throwing weapons, in which case it only works if you get the weapon back and if you get it back the whole passing weapons thing is overriden by the rules for your method of retrieval, but it is a problem should you find an edge case where it'd come up)
Making an attack is a standard action.
With a normal melee weapon, you can strike any opponent within 5 feet. (Opponents within 5 feet are considered adjacent to you.) Some melee weapons have reach, as indicated in their descriptions. With a typical reach weapon, you can strike opponents 10 feet away, but you can’t strike adjacent foes (those within 5 feet).
These rules, from the actions in combat section, are included to illustrate that regular attacks need not be made with one's dominant hand. None of them mention handedness, and they clearly allow you to attack with normal melee weapons you wield.
Normal: If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. When fighting in this way you suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand. If your off-hand weapon is light the penalties are reduced by 2 each. (An unarmed strike is always considered light.)
This section, from the Two-Weapon Fighting feat, makes it clear that the restrictions are all on the second attack, which must be made with a second weapon wielded in one's off hand. The same section is repeated nearly identically in the actual two-weapon fighting rules:
If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way.
This goes on to clarify that the same is true for double weapons:
You can use a double weapon to make an extra attack with the off-hand end of the weapon as if you were fighting with two weapons.
But not for thrown weapons:
The same rules apply when you throw a weapon from each hand.
Thrown weapons require you to actually throw a weapon from each hand, instead of wielding them, which generally requires you to have at least two attacks before getting the bonus from Two-Weapon Fighting, and to make at least one of them with your primary hand and another with each off hand. If you only have one hand you just need the one attack (since that then fulfills the 'each hand' requirement), but there is only one race with exactly one arm that I'm aware of. A ravid with Quick Draw or a returning weapon could potentailly make use of two-weapon fighting with its only hand, but that's about it.
Lastly, on page 311 of the 3.5 Player's Handbook, under the entry for 'off-hand', it states:
A character's weaker or less dexterous hand (usually the left). An attack made with the off hand incurs a -4 penalty on the attack roll. In addition, only one-half of a character's Strength bonus may be added to damage dealt with a weapon held in the off hand.