This is a question that comes up a lot, yet has never received any definitive answer.
Humans, and near-Humans
"Humans", and "near-Humans" (ancient evolutionary offshoots, such as the "Arkanians" or "Chiss" or "Zeltrons") are the most numerous 'alien' in Star Wars, by far, but which species after them comprise the most numerous in the Galactic Republic and Galactic Empire?
Candidate 1: "Duros", and "near-Duros"
Circumstantial Evidence: Many reference sources say something like this "Duros discovered hyperspace travel around the same time as Coruscant and Corellia, and are one of the earliest trading partners of those worlds, they are widespread, being enthusiastic spacers." They were one of the founding members of the Old Republic. In addition, one of their offshoots, the Neimodians, are themselves very powerful, having numerous colonies like Cato Neimodia. Seeing as how they are seen across the Cantinas and streets of the galaxy, there is strong circumstantial evidence that the Duros are the most numerous citizens of the Republic after humans.
Candidate 2: "Twi'leks"
Circumstantial Evidence: Although the Twi'lek homeworld of Ryloth is relatively barren, there seems to be a massive diaspora of Twi'lek's in the galaxy. I would speculate that ancient slave-trading dispersed them widely in Republic and Hutt space, and that descendants of these slaves form a huge part of galactic society, and also that poor migrants left Ryloth in vast numbers in ancient times for the better life found in the Republic. Somebody conducted an analysis of Wookieepedia and found that more named Twi'lek characters exist than any other alien species. Visual evidence suggests they are very very common, perhaps turning up more often than any other alien in scenes of everyday life. It's also possible that some species comprise near-Twi'leks, as others like Togruta have similar features.
Candidate 3: "Gran"
Circumstantial Evidence: Again, this is mainly going on visual evidence, but the Gran species seems to be quite common in the Republic and in Hutt space. In Star Wars: Dark Forces II, Gran thugs seem to be very numerous on city worlds like Nar Shadaa. There are famously three Gran in the Galactic Senate representing Malastare - a planet which is not even the Gran homeworld, but was homeworld of the Dug species, and was taken over by the Gran in an act of colonialism.
Candidate 4: Insectoid and hive species, such as "Geonosians"
Circumstantial Evidence: Hive planets like Geonosis often had populations of hundreds of billions (compared to only a few billion on Corellia or Duro, and not to be compared with huge city-worlds). They were implied to be some of the galaxy's key industrial producers, making consumer products, and arms for the Clone Wars for example. They may have been very numerous, but also very isolated to their home worlds, hence why they are uncommon sights in the wider galaxy.
Conclusion:
My completely non-canonical guess would be that Duros and Twi'leks are the most numerous citizens of the Republic and Empire after humans. Do you have other suggestions? Quarrens, Bith, Ithorians, Rodians, etc, seem to be quiet common too (as seen in The Clone Wars, Episode 2.11 - "Lightsaber Lost", where they comprise most aliens seen in Coruscant's streets).

Humans, and near-Humans
"Humans", and "near-Humans" (ancient evolutionary offshoots, such as the "Arkanians" or "Chiss" or "Zeltrons") are the most numerous 'alien' in Star Wars, by far, but which species after them comprise the most numerous in the Galactic Republic and Galactic Empire?


Candidate 1: "Duros", and "near-Duros"
Circumstantial Evidence: Many reference sources say something like this "Duros discovered hyperspace travel around the same time as Coruscant and Corellia, and are one of the earliest trading partners of those worlds, they are widespread, being enthusiastic spacers." They were one of the founding members of the Old Republic. In addition, one of their offshoots, the Neimodians, are themselves very powerful, having numerous colonies like Cato Neimodia. Seeing as how they are seen across the Cantinas and streets of the galaxy, there is strong circumstantial evidence that the Duros are the most numerous citizens of the Republic after humans.

Candidate 2: "Twi'leks"
Circumstantial Evidence: Although the Twi'lek homeworld of Ryloth is relatively barren, there seems to be a massive diaspora of Twi'lek's in the galaxy. I would speculate that ancient slave-trading dispersed them widely in Republic and Hutt space, and that descendants of these slaves form a huge part of galactic society, and also that poor migrants left Ryloth in vast numbers in ancient times for the better life found in the Republic. Somebody conducted an analysis of Wookieepedia and found that more named Twi'lek characters exist than any other alien species. Visual evidence suggests they are very very common, perhaps turning up more often than any other alien in scenes of everyday life. It's also possible that some species comprise near-Twi'leks, as others like Togruta have similar features.

Candidate 3: "Gran"
Circumstantial Evidence: Again, this is mainly going on visual evidence, but the Gran species seems to be quite common in the Republic and in Hutt space. In Star Wars: Dark Forces II, Gran thugs seem to be very numerous on city worlds like Nar Shadaa. There are famously three Gran in the Galactic Senate representing Malastare - a planet which is not even the Gran homeworld, but was homeworld of the Dug species, and was taken over by the Gran in an act of colonialism.

Candidate 4: Insectoid and hive species, such as "Geonosians"
Circumstantial Evidence: Hive planets like Geonosis often had populations of hundreds of billions (compared to only a few billion on Corellia or Duro, and not to be compared with huge city-worlds). They were implied to be some of the galaxy's key industrial producers, making consumer products, and arms for the Clone Wars for example. They may have been very numerous, but also very isolated to their home worlds, hence why they are uncommon sights in the wider galaxy.
Conclusion:
My completely non-canonical guess would be that Duros and Twi'leks are the most numerous citizens of the Republic and Empire after humans. Do you have other suggestions? Quarrens, Bith, Ithorians, Rodians, etc, seem to be quiet common too (as seen in The Clone Wars, Episode 2.11 - "Lightsaber Lost", where they comprise most aliens seen in Coruscant's streets).
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