First, let's identify what exactly you mean by this question, because there are two things that make the Roman legionaries potent as compared to their peers. For the first, part, clearly you are interested in the arming of the infantry. Late Republican and Principate legionaries were pretty much universally heavy swordsmen. Did other states utilize similar arming of their troops? With certainty. While there are stories of Gallic men fighting naked, for the most part the 'Barbarian' tribes, at least the front line warriors, were actually comparatively well armed. Metalworking was one of the biggest strengths of the tribes runing from the Danube to Iberia. Heavily armed and armoured infantry would not have been an uncommon sight among Rome's enemies to the West and North. Even the Greek troops could be armed as such. So the common fantasy that Rome was always more well equipped than its enemies is obviously out to lunch here. That said, in the East, such as Armenia or Parthia, infantry would have generally been much lighter armed, as mobile forces backed by a huge (as compared to the Romans) amount of cavalry was their strength.

The second aspect then is organization. And here is where the Romans shined. I am not aware of any state that attempted to replicate the type of organization, training, and most of all, battlefield discipline, that the Roman's demonstrated. While Gallic warriors could very often be very heavily armed, their method of attack was disjointed at best. Battlefield discipline is probably one of the most important aspects of all this as well, and a great deal of that comes from experience. When compared to their foes, the average Roman soldier during this period was not necessarily better armed, but they were much more experienced. A full time soldier for ten years versus locally raised militia, or men drafted for a war, is incomparable. The importance of that experience cannot be overstated. There is more than one Roman historian that has argued that the reason Rome was able to expand so quickly in the 2nd century BCE was because their legions were so experienced following the Second Punic War and Macedonian War. Indeed, some might argue this was the Roman army at its height: a force of professional soldiers that were on near constant campaigns over the course of decades. This is something that is simply not possible to copy. Indeed, let's go to the Civil War between the Triumvurs and the the Liberators. The former utilized an army of hardened veterans from Caesar's legions, as well as greener but still battle-hardened troops who had fought at Mutina. Compare that to the on paper formidable numbers of the later. Although there were almost certainly veterans of Pompey's legions interspersed in their forces, the majority were newly raised legions from the provinces, men who were trained and armed in the Roman style by Romans, but who lacked the experience. This played out over and over in fact when Romans fought Romans. Almost every civil war not decided on the battlefield was won with experience of the troops. And one cannot simply copy that.

EDIT: For those looking for sources on the Roman Army, Goldsworthy is my personal favourite. The Complete Roman Army and Roman Warfare are where most of this was pulled from, as well as Mackay's The Roman Empire. In terms of the thesis that Rome's experience was key in their expansionary success, that line is pulled directly from The Complete Roman Army.

The Dacians replicated a legionary force and used it against the Roman Empire. It was such a good fighting force that it was incorporated into the Legions after Dacia was defeated.

-- Edit: As Marsmjd pointed out below it was the Galatians, not the Dacians. Though Dacia did become a haven for roman exiles and enemies until Trajan's conquests.

The Seleucids used them, yes. I think this is what you're looking for.

http://en.wikipedia.org/wiki/Argyraspides

However the corps of men described by Polybius as being armed and dressed in the 'Roman fashion' numbered 5,000, and Bar-Kochva suggests that these men, who are described as being in the prime of life, might have also been a division of the Argyraspides, putting the number of the corps back up to 10,000 strong.

Kind of. The Renaissance saw not just the rekindling of classical political, cultural, artistic and scientific ideas but also military ideas too. In 1534 Francis I of France reorganised his armies into 'legions' of 6000 split into companies of 100. Although he was probably the one to most directly imitate Rome most rulers reorganised their armies in a similar way. Military works such as Vegetius' De Re Militari were translated into the vernacular and applied to the armies of the time. This obviously didn't produce exact copies of the Roman legions, or anything close, but renaissance armies reflected principles of methodical organisation, discipline and drill which would have been more familiar to a Roman than a Medieval general.

Source: Thomas Arnold, The Renaissance at War

Others did try to copy the Roman legions, you had pseudo-legionaries under the Seleucids, the Numidians, and I think Bithynia. None of them seemed particularly successful to become a permanent organization.

But don't be fixated on the organization of the legion, it wasn't the end-all-be-all in and of itself. In addition to losing (very badly) in some battles against Hannibal, several armies got themselves badly thrashed by the Gauls and Germans in the early Late Republic.

But there are several factors that made it so successful, in no particular order:

• Numbers. Even if you defeated one army another would follow. And another and another if necessary. The Romans were really good at integrating defeated peoples into their empire and so with every victory their manpower pool grew.
• Generalship. We tend to downplay the efforts of generals and emphasize the presence of centurions, often equating them with NCO's. While Rome did get some atrocious commanders, they also had an awful lot of very good ones. Plus these would have come in having apprenticed under their fathers, uncles, of family friends - thus the odds of getting absolute duds would be lower.
• Command and control. The legion's makeup and the general's role of staying behind the front line and managing the battle gave Rome an advantage in battles. A Hellenistic general fighting along with his troops would definitely be good for local morale but he won't be able to actually 'general' his whole army.
• Combat system. The scutum, gladius, and pila combined with an aggressive Roman attitude was remarkably effective in dealing with a variety of enemies.

The Roman legion XXII Deiotariana was created by merging together two "legions" raised by the Galatian king Deiotarus. These two legions imitated the Roman style of organization and combat so well that the emperor Augustus chose to incorporate them into Rome's armies rather than disband them after their defeat.

Source: Legionary: The Roman Soldiers Unofficial Manuel by Philip Matyszak.

A similar question: how were Armenian armies during the Roman imperial period organized and equipped? I ask this in this thread because I suspect that the answer is that they mimicked the Roman legions.

If you want one example, the Hellenic Seleucid Army has been recorded as copying the Roman legion, although the Seleucid troops adorned themselves in silver armor instead of the Roman red and were referred to as "Silver Shields"