[RPG] How do attack rolls work with two-weapon fighting

You can wield & attack with two weapons without any special class feature or feat.

Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):

Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.

Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.

The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.

Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.

A long-running Pathfinder campaign in which I was involved that ended when characters were level 14 included—at one point or another—a half-orc inquisitor, a human barbarian, a human cleric, and a vanara druid. The campaign had the PCs as police detectives in the Magic: The Gatering setting Innistrad. Adventures involved a great deal of investigation and mundane legwork plus plenty of we've-discovered-your-evil-plot-so-it's-over! sort-of reveals in addition to beating the crap out of a lot of evil outsiders and undead.

This list is in no way comprehensive and instead limited to items I've seen used in play in an 18-month-long campaign that had a significant combat encounter only about every other session. I've further limited the list to only the items a fighter could employ.

• The half-orc inquisitor wore the bracers of the falcon's aim (4,000 gp; 1 lb.) that grants a +3 competence bonus on Perception skill checks in addition to other effects; the jingasa of the fortunate soldier (5,000 gp; 3 lbs.) which, while a combat item, I'd be remiss were I not to recommend; and the mushroom vest (500 gp; 3 lbs.) that lets the wearer treat falls as 20 ft. shorter, which, admittedly, isn't the boots of the cat (1,000 gp; 1 lb.), but the boots slot is premium real estate in Pathfinder. When needed, he'd doff the vest and don a robe of infinite twine (1,000 gp; 1 lb.). I'm pretty sure he also wore the heavyload belt (2,000 gp; 3 lbs.) and the muleback cords (1,000 gp; 0.25 lb.) so that the weight he could lift, drag, or push was measured in tons. At later levels, he discovered the gloves of reconnaissance (2,000 gp; 0 lbs.) and cried a little, regretting he'd not been wearing them all along, then never took them off.
• The human barbarian with the drunken rager archetype had several utility magic items so that, despite not casting spells, he could feel useful during investigative portions of the session. These included the cloak of the hedge wizard (transmutation) (2,500 gp; 1 lb.) that cost 50% more because it was combined with his cloak of resistance +3 (4,500 gp; 1 lb.), a wayfinder (500 gp; 0 lbs.) with an ioun stone (clear spindle) (4,000 gp; 0 lbs.) so that together they resonated with an immunity to mental control as per the spell protection from evil, the ioun stone (eastern star) (4,000 gp; 0 lbs.) so as to be able to read and understand all languages, the noble's vigilant pillbox (3,600 gp; 0 lbs.) because of its detection abilities, the ring of the sophisticate (11,000 gp; 0 lbs.) so he could find a bar anywhere, and, later in the game, the expensive drinking horn of bottomless valor (24,000 gp; 2 lbs.) that was useful also as a source of endless booze so as to freely liquor up suspects and witnesses to get them to talk.¹
• The human cleric's player's first big purchase for any of his characters is almost always the hat of disguise (1,800 gp; 0 lbs.), which is probably near the top of almost anybody's Versatile Magic Items list. Looking back, I would suggest now that he buy first—and later pay the extra 50% to combine the hat with—a cap of human guise (800 gp; 0 lbs.) because it's cool, too.
• The vanara druid's elephant animal companion wore a belt of the weasel (10,000 gp; 1 lb.) so that the elephant could more easily enter people's homes during investigations. (Shockingly, folks were still uncomfortable having an elephant in their homes.) However, the party soon realized the belt's utility value, and the belt ended up as often on a party member as the elephant. If the belt seems overpriced, then reread the universal monster ability compression that the belt also grants: it's awesome ("You can squeeze through a space how little exactly?"). The vanara druid herself wore a ring of eloquence (3,500 gp; 0 lbs.) but custom made so as to include, instead of its typical languages, the four elemental languages so that she could communicate with her summoned allies.

In retrospect, we weren't using several items that we probably should've been using. In addition to several ioun stones (cracked orange prism) (1,000 gp; 0 lbs.) each for the casters because more 0-level spells is incredible, I'd want now, for instance, the headband of intuition (7,000 gp; 1 lb.), at least one insistent doorknocker² (5,000 gp; 2 lbs.), and the treasure hunter's goggles (6,400 gp; 0 lbs.).

¹ The drinking horn supplemented his low-level purchase, the tengu drinking jug (1,000 gp; 2 lbs.). He'd tired of the one gallon of sake or plum liquor ("Tastes like pruno") the jug could create each day, even though it was sort of free.
² Two doorknockers are better so that you can make a door then make another door on the other side to recover the first doorknocker, but even just one's great for a case of the lookarounds.