The only way to fight with dual-wielded weapons is by equipping a normal one-hand weapon in the main weapon slot and an "offhand" weapon in the offhand/shield slot. These offhand weapons are separate items from the regular weapons, each corresponding to an existing one-hand weapon. Each offhand weapon can be obtained in the same way as its corresponding regular weapon.
Almost every one-hand weapon has a corresponding offhand weapon. A notable exception is the Abyssal Whip, which apparently had animation issues. One other oddity is that dragon claws, previously two-hand weapons, are now separately obtainable as claw and off-hand claw.
As far as I can tell, the only real difference that comes from the extra weapon is the extra DPS.
Almost all of this information comes from this reddit thread I posted.