For whatever reason, I've never been a big fan of dual-wielding when I could be using a massive two-handed weapon. In World of Warcraft, Enhancement Shamans briefly used such big weapons during vanilla, but with the release of Burning Crusade and for the last 15 years, they've dual-wielded one-handed weapons.
In D&D 5th Edition, dual-wielding can be a decent option, but it runs into some pitfalls.
Here are the rules on dual-wielding, or, as D&D calls it, two-weapon fighting:
You can only do this if the weapons in both hands have the Light property. Then, if you take the attack action with the weapon in your main hand, you can make a single attack with the weapon in your off hand as a bonus action. This attack does not add your Strength (or Dexterity) modifier to the damage.
Ok, so let's take a look at this:
Say you're a dual-wielder and you pick scimitars as your weapons of choice. You have one in each hand, and a +3 to Dexterity (scimitars being finesse weapons, and this feeling like a kind of dexterous fighting style - though you could easily do this with handaxes if you're a strength-based character. You could also go for shortswords if you prefer piercing damage, and a few other kinds of weapons if you are ok with a lower damage die).
We're going to assume for the sake of argument that you are playing a martial class that gets the Extra Attack feature at level 5. Dual-wielding actually works quite well for Rogues, largely because most of their damage comes from Sneak Attack, but we're going to set them aside. So, we'll be talking about Fighters, Paladins, Barbarians, Rangers, and some Artificers and Bards (Monks I'm going to leave out because they essentially get to "dual-wield" their weapon and their unarmed strikes).
For a level 1 Ranger or Paladin, dual-wielding actually works pretty well at 1st level. If you figure that a paladin could be wielding a maul or greatsword, which does 2d6 + their Strength modifier (+3 in our hypothetical,) dual-wielding actually gives them the same - 1d6+3 for the main hand and 1d6 for the off hand.
Fighters, who get a Fighting Style at level 1, could pick up two-weapon fighting, which allows you to add your modifier to the damage of an off-hand strike. Thus, a Fighter who was specced into this could do 1d6+3 and then another 1d6+3, giving an average of 13 damage. Compare that same fighter going for Great Weapon Fighting and a Maul, and they're doing an average of 8.333 + 3, or 11.333.
Barbarians don't get fighting styles, but they do get Rage. At level 1, their Rage bonus is +2 to damage, and so while they're adding +5 to a main hand attack, they're still adding 2 to that off-hand strike. Thus, they can do effectively 2d6 +7, which comes out to about 14 damage per turn... but only after the first turn.
See, the big downside is that this costs more than just your action to attack - you also need your bonus action.
Rangers can pick up two-weapon fighting, and it looks pretty similar to the Fighter. Indeed, they don't get great weapon fighting, so it seems the expectation is that they will go with two-weapon fighting if they plan to be in melee.
Paladins don't actually get access to this fighting style. When we get to higher levels, though, you might actually be encouraged to try it nonetheless.
As we get to higher levels, this starts to compete with other features. A Ranger with Hunter's Mark will need to use their bonus action to place and then move their Hunter's Mark (or Favored Foe, though I think that is the one alternate rule from Tasha's that doesn't really live up to what it should be).
The other place this starts to break down is at level 5, when we get Extra Attack. This does allow us to make another main hand attack, but we're still only making one off-hand attack.
Before we continue, though, here's something you might not know: You can actually make your two attacks with different weapons, one in each hand. There's nothing that prevents you from holding two heavier weapons and fighting with them (though you can't attack with two-handed weapons when you're only holding them in one hand). But you can't then make the bonus action attack.
That aside, let's look at things at level 5.
With a Fighter, the Great Weapon fighting style simply doubles its potential damage at level 5. We can assume a +4 to Strength or Dex at this point. So our great weapon guy is doing 8.333+4, all time 2. So, we get 16.667+8, giving us about 24.667 damage per round. Dual-wielding, we get (1d6+4)x3. That comes out to an average of 22.5 - a little behind.
The Ranger is probably not using a two-handed weapon (at least one that's not a ranged weapon). Their dual-wield damage is the same as the Fighter's. However, a feature like Hunter's Mark benefits greatly from getting multiple attacks. While they can't pull this off on the first turn (where they're doing a mere 4d6+8 if they hit with both attacks - which still comes out to 22) on subsequent turns if the target's still around, you would get 6d6+12, or 33 - not bad.
Our Paladin is going to fall pretty fall behind, their main hand attacks doing a total of 2d6+8 (so about 15) and then just 1d6 on the off-hand, adding 3.5 damage, for 18.5 total. Again, if they go for a two-hander and great weapon fighting, like the Fighter, they get 24.667, which is quite a bit more.
Barbarians at this level are doing 2d6+12 with the two main hand attacks, or about 19, and then 1d6+2 with the off hand, or about 5.5, giving us about 24.5. Compare that with a maul and you'll get 4d6+12, which is 26. Better, but honestly not super far ahead. However, again, like the Ranger, we need a turn to get Rage going (though unlike the Ranger, we don't have to switch the damage bonus to a new target with subsequent bonus actions).
Let's jump to tier 3, aka level 11. Our Strength/Dexterity is now up to +5, and we'll imagine we've gotten +1 weapons by now.
At this point, the Fighter has three attacks as part of the attack action. As such, the single off hand attack gets further devalued. With Great Weapon fighting and a maul, 3 attacks, +5 strength, and +1 weapons, we're looking at 6d6+18, and with the fighting style, that comes out to 25+18, or 43 damage per round. Dual-wielding, we're now at 3d6+18 for main attacks, or 28.5 for our main hand, and 1d6+6 for our off-hand, which is about 9.5, giving us 38 damage per round.
The Ranger at this point actually might benefit less from dual-wielding because there's a good chance they're using concentration for something else. Summon Fey, for instance, will out-damage Hunter's Mark (and also require no bonus action to send after other targets). If we assume they are still using Hunter's Mark, though, we basically just add two damage per attack thanks to their +5 to Dex and the +1 weapons, so we're up to 39 - still good, but other classes are catching up. By contrast, for example, Summon Fey would leave our normal attacks at just (1d6+6)x3, or 28.5, and then 2d6+6 from the fey spirit cast at 3rd level, which is on average 13 damage, giving us a total of 41.5 damage per round.
Now, here's where things get kind of interesting for the Paladin. Officially, Great Weapon Fighting only applies its bonus to the weapon damage, and not things like Divine Smite or the confusingly named Improved Divine Smite. So, our GWF Paladin is doing 2d6+1d8+6 damage with each attack, the 2d6 getting the Great Weapon Fighting bonus. This means we've got 8.333+4.5+6, or a total of 18.833 damage per hit, and thus 37.667 damage total. Now, if we dual-wield, things start to get kind of interesting. Each hit is now adding 1d8: so our two main hand attacks are 1d6+1d8+6, which comes to about 14. Two attacks means 28 damage, and then our off-hand is going to do 1d6+1d8+1, or about 9. Thus, dual-wielding we get 37, which is just barely below our two-hander style. You could make the argument that, because Paladins are always fishing or crits to use Divine Smite, having one more attack per round could make that a lot more likely (two attacks give you about 9.75% to crit, while a third will make that nearly 14%).
In fact, we can then make this a bit crazier with the Paladin if we take a look at the spell Spirit Shroud. This adds 1d8 radiant, necrotic, or cold damage to anything you hit with an attack within 10 feet of you. Again, you've got to spend a bonus action activating it, but this adds a total of 3d8 damage to your dual-wielding situation - about 13.5, versus just 9 if you're using a two-hander. So, two-hander goes to 46.667 while dual-wielding goes to 50.5. Dang.
Now, Barbarians are going to get a +3 Rage bonus now. With a Maul, we're looking at two attacks that do 2d6+5(strength)+3(rage)+1(weapon). So, each attack does about 16 damage, so our total damage is 36 - slightly behind, but not by a huge amount. Dual-wielding, though, we're now looking at two attacks that do 1d6+9, for a total of 25 damage, and then an off hand that does 1d6+1+3, or 7.5, for a total of 32.5.
Where this does get a little complicated (and beyond the math that I really want to get into) is that the Barbarian also gets Brutal Critical at these levels, which lets you add more dice when you get a critical hit. While it's not until very high levels that this winds up making a Greataxe better than a Maul or Greatsword, the odd upshot is that basically, for a Barbarian, having fewer dice per weapon is better. A At level 9, for instance, a Maul critting becomes 5d6, while a Greataxe becomes 3d12 - in other words, 17.5 versus 19.5.
While I haven't calculated it here, I believe that, by having each attack represented by a single die (plus modifiers,) dual-wielding I believe makes Brutal Critical more helpful than simply swinging a maul or greatsword.
Essentially, the key with dual-wielding as it works is that it's best for classes that have extra things that happen when they land hits. A fighter, for example, is already about making tons of attacks, so a single extra one is really not that big a deal. But for classes like the Paladin and Barbarian, who, at higher levels, can add a lot of damage to each hit, it starts to look a lot more appealing.
I would say, though, that the fighting style itself is not necessarily the be-all end-all of whether you should consider dual-wielding. The paladin doesn't have access to it, but winds up doing insane damage (admittedly with a relatively high-level spell). The fighting style is only ever going to add 5 damage per round. And if you really want it, you could take the Fighting Initiate feat to get it anyway (though paladins are going to probably want to max out strength, constitution, and charisma).
Another feat of note if you're considering this weapon load-out is, of course, Dual-Wielder. This does a few things: first, it allows you to dual-wield weapons that are not light - so you could have two longswords or warhammers. It also lets you draw them as a single item interaction... something that I think most DMs would let you do anyway. Lastly, while using those two weapons, you get +1 to AC, which is actually a pretty nice compromise to make up for not using a shield. Getting non-light weapons is still going to probably limit you to 1d8 weapons, meaning on average you're only adding about 3 damage per round (3.5 versus 4.5 average rolls). However, it also means that if your party comes across magic weapons, you'll have more options.
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