
It should also be noted there are distinct differences between Eorzean firearms, Machinist firearms, and Garlean firearms...
"Common" Eorzean firearms generally tend to be in the style that you would see in on the hip of certain Yellowjacket and Lominsian NPCs. They have a less intricate look and tend to look like real world black powder pistols, I personally refer to them as firesand pistols. They come across as single shot, pellet-based weapons (if their design is assumed to mirror their real world counterparts) meant as a last resort or for when you really need to drop someone or something fast so that the considerable reload time doesn't hinder further conflict. This is of course barring cutscene which show the Admiral firing multiple rounds from her double-barreled pistol which can be seen as mostly cinematic. Eorzeans also have muskets as evidenced by the unfortunately unused Musketeer's guild in Limsa. These have longer range, but still would likely fall under the single shot style of weapon.
Garlean firearms tend to come in a couple of flavors. They appear to have developed revolvers even before the advancements in magitek (see also the Pre-Imperial Garlean Revolver which drops from Gubal Library normal mode). As such revolvers have fallen into use in just about every aspect of their firearm technology. The standard issue Garlean gunblade has a revolver cylinder for ammo for instance. This ammo I believe falls under more modern cartridge based ammo rather than pellet rounds. Garlean characters (or those from Garlean provinces) may be able to get their hands on revolvers if such a thing is explained. While we haven't visibly seen any Garlean long-barrel rifles, it stands to reason that if Eorzea has them, it's likely Garlemald has developed something similar.
They also have Magitek arms which only seem to be used by upper echelon Garleans if not mounted on their mobile weapons. These tend to fire blasts of charged and volatile ceruleum with the intent of explosive impact damage rather than penetrative. The likelihood of a random Eorzean/Garlean having such a weapon is slim to none as we've only ever seen Legatus-level characters with these weapons (Gaius' wrist-mounted magitek cannon, Regula's magitek greatsword, etc)
Lastly and most recently we have Machinist guns. This are given freely to those who would sign up with the Machinists at Skysteel Manufactory. These weapons tend to be modified arms which (one would assume) fire cartridge based rounds rather than the pellet rounds of older firesand weapons. This is evidenced from the fact that one can load multiple rounds into the weapon at once and they can be fired in quick succession. This also goes without mentioning the Aetherconverter which Machinists make use of. The converter uses your own aether to empower your weapon with aether-based rounds which do not require reloading but do not hit quite as hard as a physical round. Combining this with physical rounds actually increases the damage output of your weapon such that you'll be doing even more damage than if you were firing physical rounds alone as well. Given that the aetherconverters are a creation of the Skysteel Manufactory though...there are no real exceptions to how to get one other than becoming a Machinist.
Now with all that stated I know I didn't exactly quote any lore openly so if there is anything that contradicts what I'm saying, please do correct me, but otherwise I hope this is helpful.
"Common" Eorzean firearms generally tend to be in the style that you would see in on the hip of certain Yellowjacket and Lominsian NPCs. They have a less intricate look and tend to look like real world black powder pistols, I personally refer to them as firesand pistols. They come across as single shot, pellet-based weapons (if their design is assumed to mirror their real world counterparts) meant as a last resort or for when you really need to drop someone or something fast so that the considerable reload time doesn't hinder further conflict. This is of course barring cutscene which show the Admiral firing multiple rounds from her double-barreled pistol which can be seen as mostly cinematic. Eorzeans also have muskets as evidenced by the unfortunately unused Musketeer's guild in Limsa. These have longer range, but still would likely fall under the single shot style of weapon.
Garlean firearms tend to come in a couple of flavors. They appear to have developed revolvers even before the advancements in magitek (see also the Pre-Imperial Garlean Revolver which drops from Gubal Library normal mode). As such revolvers have fallen into use in just about every aspect of their firearm technology. The standard issue Garlean gunblade has a revolver cylinder for ammo for instance. This ammo I believe falls under more modern cartridge based ammo rather than pellet rounds. Garlean characters (or those from Garlean provinces) may be able to get their hands on revolvers if such a thing is explained. While we haven't visibly seen any Garlean long-barrel rifles, it stands to reason that if Eorzea has them, it's likely Garlemald has developed something similar.
They also have Magitek arms which only seem to be used by upper echelon Garleans if not mounted on their mobile weapons. These tend to fire blasts of charged and volatile ceruleum with the intent of explosive impact damage rather than penetrative. The likelihood of a random Eorzean/Garlean having such a weapon is slim to none as we've only ever seen Legatus-level characters with these weapons (Gaius' wrist-mounted magitek cannon, Regula's magitek greatsword, etc)
Lastly and most recently we have Machinist guns. This are given freely to those who would sign up with the Machinists at Skysteel Manufactory. These weapons tend to be modified arms which (one would assume) fire cartridge based rounds rather than the pellet rounds of older firesand weapons. This is evidenced from the fact that one can load multiple rounds into the weapon at once and they can be fired in quick succession. This also goes without mentioning the Aetherconverter which Machinists make use of. The converter uses your own aether to empower your weapon with aether-based rounds which do not require reloading but do not hit quite as hard as a physical round. Combining this with physical rounds actually increases the damage output of your weapon such that you'll be doing even more damage than if you were firing physical rounds alone as well. Given that the aetherconverters are a creation of the Skysteel Manufactory though...there are no real exceptions to how to get one other than becoming a Machinist.
Now with all that stated I know I didn't exactly quote any lore openly so if there is anything that contradicts what I'm saying, please do correct me, but otherwise I hope this is helpful.