An image was recently posted in r/ArtefactPorn of a mosaic at the entrance to a Roman-era (1st-2nd century AD) bathhouse in Timgad, Algeria. The mosaic depicts two messages, one of which was "BENE LAVA" (Have a Good Bath) and the other presumably "SALUM LAVISSE" (Hope You Had a Nice Bath), forming a welcome and goodbye sign, as each was flipped to read correctly for readers in either direction along the long edge. Essentially it looked like a modern bathmat.
In the centre are two sandals similarly oriented, suggesting that they guided viewers in the direction of travel (coming and going). On either side, the sandals on the left represented the viewer's direction (the straps are to the front).
What do we know about directing pedestrian traffic in the Roman Empire? Could it just have been an accident that the sandals were oriented this way in the mosaic or was it practice that Roman society preferred to walk on the left? How does this square with driving on either side of the road in the Roman Empire, given that over the years there have been two (rather than one) dominant systems in the world?
In 1998, a Roman quarry was discovered near the English town of Swindon. For centuries, the local Portland limestone had been cut and shaped into blocks for the needs of this quiet corner of Roman Britannia. And for centuries, heavy four-wheel carts had rattled in and out of the works, scoring the stone. By the time the quarry was abandoned, two sets of well-defined ruts had formed. One set - on the left, looking out from the quarry - was much deeper than the other. This, the archaeologists realized, reflected the traffic pattern on the quarry road. Year after year, carts loaded with stone had exited by the left-hand ruts, and returned empty on the other side.
The traffic pattern in the Swindon quarry is sometimes cited as evidence that the Romans habitually drove on the left. It is, of course, nothing so definitive. We can say with bedrock certainty (so to speak) that carts at the Swindon quarry kept to the left. But we cannot assume that this was a universal rule, or that pedestrian traffic followed a similar pattern.
Typically, wheeled traffic kept to the center of a Roman road. Often, in fact, it had no choice. The best evidence, as so often, comes from Pompeii, where the roads of a bustling Roman town have been preserved more or less intact. As I discuss in an older answer, vehicular traffic was limited on the streets of Roman cities. It was also carefully channeled. In Pompeii, at least, most of the streets were one-way, and relieved only by occasional turn-offs and parking spots (to judge from the municipal laws of a Roman town in Spain, there were hefty fines for blocking traffic). Other cities were provided with wider, two-lane streets, whose traffic must have been guided by regulation, or at least convention. But there is, to my knowledge, no evidence for a universal right- or left-hand rule.
Pedestrian traffic seems to have been fairly anarchic. Although there must have been local conventions for walking through narrow areas, these rules were unwritten, or are at least not preserved. Some large buildings, above all the great imperial bath complexes of Rome, were designed for continuous circulation in both directions. Here again, however, we have no indications of a general rule for foot traffic keeping to one side or the other.
In the absence of definitive evidence, in short, we should probably assume that local custom governed pedestrian behavior everywhere, and that the practice of keeping to the left - if it was indeed the general rule on Roman roads - was not a universal guide for walking patterns.
You can read more about Roman traffic in this interesting article, available on JSTOR.