The term "urban legend" was created to describe modern, traditional stories to be believed. Legends have always fallen into several groups, but the genre hangs together as stories that are generally intended to be believed. Not all of the stories are traditional (even if they employ traditional beliefs).

For example, if I tell someone, "you wouldn't believe what I saw last night. It was really weird and I think it was... [fill in the blank with an entity that is regarded as possible by modern standards - including, an angel, a ghost, an extraterrestrial]." That story may not be repeated so it ends with that. It is what the great folklore theoretician Carl Wilhelm von Sydow (1878-1952) called a "memorate" - a personal account that may employ traditional beliefs but does not rise to the level of becoming a traditional story by being repeated. Probably all cultures tells these short forms of stories, so we can imagine the Romans did as well.

Other legends consist of traditional stories and beliefs. In pre-industrial cultures, we generally refer to these as migratory legends. The first to catalogue these stories was the great Norwegian folklorist Reidar Th. Christiansen (1886-1971). These tended to die out with modernization. When folklorists recognized that modern society had legendary-like stories being told over distance and time, they arrived at the term "urban legend" in recognition that these stories were modern and they thrived in an urban setting. There is a flaw in the distinction implied by this name since migratory legends could also thrive in a pre-modern urban setting. In fact, the term "urban legend" is unfortunate because it implies that "our" folklore is different from that of pre-industrial people.

The only significant difference between an urban legend and the migratory legend is that the former tends to be debunked quickly, so its life is often shortened. Also, some of these legends are dependent on technology, and so when technology changes or becomes general accepted and understood, the urban legend fades. Thus, when I ask my folklore classes if anyone has heard the story of the poodle in the microwave or of the foreign tourist misunderstanding how cruise control works while driving on an LA freeway, students younger than 40 tend to have never heard the stories, while those older remember them very well. These were very popular in their time, but they quickly become obsolete and so they faded away.

Which gets us to the Roman Empire and the question of migratory legends. These would be what people in the empire would have told to be believed - the counterpart of today's urban legend. Since we have a great many sources from the period, we can recognize many stories that were captured in Roman literature that were repeated until the nineteenth century, and we can glimpse stories that may not have survived but were likely traditional for at least a while even if they did not survive.

For example, the first-century "Satyricon" includes two legends, one involving the supernatural abduction of an infant and the other a man transforming himself into a werewolf. Motifs associated with both stories are recognizable in European folklore collections from the nineteenth century. Other stories appear in primary sources involving Roman-era mythology, and these too (they were stories told generally to be believed) are often echoed in pre-industrial migratory legends.

So the answer to your question is that people in the Empire did indeed tell stories to be believed that echoed what we know today as urban legends. The traditional European migratory legend typically lasted longer than an urban legend, but it behaved in much the same way. These were entertaining stories that were thrilling to hear and imagine to be true. Skeptics in any century may ruined the fun but the genre continues to thrive.