Not exactly. Although there were instances of first-century emperors using Praetorian centurions and tribunes to arrest those accused of treason (e.g. Tacitus, Annals 11.37, 15.60), Rome's true "secret police" were the Frumentarii, small groups of soldiers detached from their legions for special service in Rome and the provinces. But since the Frumentarii in Rome were under the direction of the Praetorian Prefect, their association with the Praetorians was always close.
Originally, as their name suggests (frumentum is grain in Latin), the Frumentarii were responsible for logistical matters. By at least the beginning of the second century, however, they had assumed a series of additional functions: carrying imperial messages, collecting certain taxes (probably levied in the form of grain), and - most relevantly - serving as informers and secret police.
The Frumentarii in Rome seem to have been used largely to spy on prominent senators and members of the imperial court. According to the author of the Historia Augusta, for example, Hadrian used Frumentarii to learn the secrets of his friends and acquaintances among the Roman elite (11.4). In his Ecclesiastical History, likewise, Eusebius claims that Frumentarii were used to hunt down Christians (6.40). They were also (at least if we can believe the Historia Augusta) employed by less scrupulous emperors to carry out assassinations.
William G. Sinnigen, "The Roman Secret Service" The Classical Journal 57 (1961), 65-72
You can see why someone might say that, but I think its a bit imprecise.
A modern term for the Praetorian Guard that I think is a better fit would be a "gendarmerie". That's a military unit with civil order responsibilities. Italy's Carabinieri or Spain's Guardia Civil come to mind. The difference between those entities and the Praetorians would be that Roman unit was primarily an Imperial bodyguard.
It's still not a perfect match, but I think its much closer parallel than "secret police". The essential component of a gendarmerie is that its a military unit, typically under the military chain of command, which makes its operation much less subject to the civil regulations designed to regulate a police force.
In the case of Rome, there was a long history of laws and regulations around policing; the Praetorians weren't subject to any of that; they acted on the Emperor's orders or authority, many fewer restrictions. You also might analogize to death squads attached to various tyrants-- think of the Mano Blanco ("white hand") death squad in Guatemala, for example.
Sources:
LOOKING SOUTH: GENDARMERIES IN THE MAGHREB AND TURKEY
and especially
The Paradox of Gendarmeries: Between Expansion, Demilitarization and Dissolution
There's less material than one might like on policing in Republican and Imperial Rome-- note that the rules change substantially over time, the Augustan Reforms in particular set down rules with a new specificity. I like this article for the way that it explicitly looks at Roman and modern policing in parallel.
And on death squads:
The Political Economy of Death Squads: Toward a Theory of the Impact of State-Sanctioned Terror
As /u/toldinstone has written, the frumentarii did the bulk of spy work, not just in Rome but throughout the empire. However, as /u/toldinstone has written, they usually operated under the supervision of the Praetorian Prefect. They were effectively an unofficial wing of the Praetorian Guard, so I'd like to discuss how spy work fitted into the wider jobs of the Praetorian Guard, particularly before Trajan and Hadrian made the frumentarii the main clandestine force of the Roman Empire. It's worth noting that the frumentarii became secret police under Trajan and Hadrian, but they were a replacement for the speculatores, which was a rank in the Praetorian Guard from their inception.
One of the problems we face when studying secret operations is that they were secret. We can only guess based on the often wild stories of our ancient sources, and on peculiar circumstances that point to something clandestine.
The Guard had a lot of roles. They protected the emperor and their family, guarded imperial residences, paraded at special events, worked crowd control at special events, and did engage in spying activities. A division of the Praetorian Guard were called speculatores. Originally, these were military scouts, but since the Guard spent most of its time idle in Rome the only thing to speculate about was the activities of the emperor's potential enemies. The speculatores augusti were the speculatores of the Praetorian Guard, and as far as we can tell they were the intelligence wing of the Guard and almost nothing else. This was a high ranking position, given special privileges above most of the Guard, and were the inner circle of the emperors Galba, Otho, Vespasian, Titus, and Domitian.
The speculatores were expected to communicate execution orders, round up traitorous elements, and appear to have had limited powers over local soldiers in order to fulfil their orders. Not only to Tacitus and Suetonius attribute many such activities to the Guard, but so does The Bible (Mark 6.26-7), and Josephus noted that they were always buzzing around Vespasian and Titus. They were also messengers before the frumentarii were, and dealt with all imperial correspondence. If you wrote a letter to the emperor, you could be fairly certain that the speculatores had read it at some point.
The height of their power was in 69 AD, the Year of Four Emperors. They served as operatives and bodyguards for Galba and Otho. There was an information war in this year - Tacitus tells us that Galba recieved conflicting information from his messengers, such as how one letter said Hispania was loyal to him, but he recieved a letter the next day saying that actually the province was supporting Vitellius. Something is plainly iffy here, and it's likely that some speculatores of the Guard - the ones who conveyed these messengers to and from the emperor - were acting as double agents. When Otho seized power on 15th January 69, he had pitched his coup to the speculatores of the Guard, only to find that other officers had been trying to find Otho to pitch a coup to him. With Otho, they devised a whole plot that exploited Galba's weaknesses, complete with codewords, assembly points, and a list of pro-Galba centurions to be rounded up. In this episode, Tacitus tells us that the speculatores also handled passwords and ciphers, as one of them is identified as 'officer of the passwords'. There is simply no way the speculatores, and in fact a large portion of the main Praetorian Guard, weren't acting as secret police here.
Exactly why Trajan and Hadrian wanted to replace the speculatores with the frumentarii, especially when they do essentially the same job, isn't clear.
Sources:
Bingham, Sandra. The Praetorian Guard: A History of Rome's Elite Special Forces. IB Tauris, 2013.
Bingham, Sandra J. The praetorian guard in the political and social life of Julio-Claudian Rome. Diss. University of British Columbia, 1997.
Sheldon, Rose Mary. Intelligence Activities in Ancient Rome: Trust in the Gods but Verify. Routledge, 2004.