I heard several times that thieves and bards can be proficient in two-weapon fighting, but I don't see that anywhere in the Player's Handbook. In fact, I don't even see two-weapon fighting mentioned as a proficiency. I'm just starting to fully grasp the AD&D2E rules, so I'm oftentimes not sure where to look for specific rules.
First of all, there is no two-weapon fighting proficiency in AD&D2E (in the Core at least). You don't need any proficiency to do it. The prerequisities for doing that are in the combat section of PHB (chapter 9). There is a subsection titled Attacking With Two Weapons there.
A tricky fighting style available only to warriors and rogues is that of fighting with two weapons simultaneously.
Thus, it is stated explicitly that this fighting style is available only to warriors and rogues. Since both thief and bard are subclasses of a rogue in AD&D2E, both can use it.
I should also note that ranger in AD&D2E is a subclass of a warrior, thus can utilize the two-weapon fighting style, but ranger also has a class feature that allows him to use two-weapon fighting without regular penalties. All other classes have penalties on two-weapon fighting.