Each class has a specific weapon skill. Each Advanced Class also has a specific weapon, and off-hand, skill. For example, Smuggler Gunslingers use two blaster pistols; Smuggler Scoundrels use a blaster pistol, and a shotgun.
The weapon skill, and off-hand, are solely based on the class, and Advanced Class.
Here is a post listing the weapon (and armor) proficiency of classes, and Advanced Classes:
List Of All Classes Passive Abilities (Armor/Weapon Proficiency)