There are no answers to this question after so long because it is very difficult to answer with any certainty. In no ancient source is there any mention (to my knowledge) of the streets outside the building. We can make some educated guesses, however. First, the building itself: there were 80 archways around the lowest floor of the outer facade, all of them leading to the seats and corridors of the interior. They were numbered so that you could (or should) use the correct entrance. It is possible that these were utilized as makeshift vendor stalls when the games weren't on, but I don't know for sure. If anything, probably something along the lines of the trinket sellers in modern Rome who spread out their stuff on blankets and run for their lives when the Carabinieri come to chase them away. Beyond the ground-level archways, 18 meters out in the "sidewalk" (if you could call it that), were set stone bollards (you can still see a few today). Nobody knows what these were supposed to do: maybe establish a boundary for religious purposes, or create ticket gates, or provide anchors for the awnings above (or all three). Second, a quick look at the immediate surroundings of the amphitheater through time:

80 CE - The amphitheater is officially finished and inaugurated by the Emperor Titus. The building sits in/atop an open area of what had been Nero's famous, sprawling Domus Aurea ("Golden House"). At this time, the 120-foot colossal bronze statue of Nero still stood in its original position to the west of the Colosseum, though now altered slightly to represent the god of the sun. This area had been a massive porticoed vestibule for the Golden House previously, but I'm not sure how exactly it looked by 80. Nero's successors were embarrassed by the Golden House and wasted no time in stripping, destroying, or burying it. It is very possible that some of the porticoes framing the colossal statue were left intact. If so, then the area immediately to the west of the Colosseum would have been a broad area, with shaded walkways for strolling and possibly commerce. To my knowledge, no remains of that area exist today (see below). Regardless of merchant presence or not, it would have been an area of busy foot traffic, as just steps further to the west was the Forum, still the heart of the city in this period. The Colosseum sat in the spot where the Palatine Hill (to the west and south) and the Esquiline Hill (to the north and east) came together. In 80, the Emperor Titus was still constructing his Baths just across the "street" to the northeast of the Colosseum. They would be completed in 81. After that, the area was probably busy quite often, and positively packed when the games were on.

95 CE** - In the 90s CE, the Emperor Domitian constructed a large fountain just west of the Colosseum called the meta sudans ("sweating turning-post"). Recent excavations around and beneath it have revealed the razed remains of colonnades which had once faced out onto the great Lake of Nero's Golden House, above which the Colosseum was built. This fountain marked an important spot in the city, where many different neighborhoods met in their corners, and also where the Triumphal processions made a turn onto the Sacred Way into the Forum. It was a meaningful spot and the sort of place the Romans liked to monumentalize with architecture.

141 CE - The area around the Colosseum changed a lot in the early decades of the 2nd century CE, and was probably closed for construction periodically either partially or in full. In 141, the massive Temple of Venus and Roma was completed to the west, in the space formerly occupied by the colossal statue of Nero-Sun. The statue had been moved from its spot and placed closer to the Colosseum to make room for this massive project (thereby eventually lending its name to the amphitheater). This temple was situated on a massive stone platform, burying whatever architecture was still remaining of the Golden House beneath. This building must have changed the "feel" and "flow" of the Colosseum neighborhood quite a lot, and must have certainly funneled traffic to its north and south (towards the Baths of Titus and towards the fountain, respectively). Speaking of baths: the massive Baths of Trajan had been completed in 109 CE, looming just to the north of those of Titus.

This completed the "neighborhood" of the Colosseum in the High Empire. Constantine eventually added his famous triumphal arch just to the south of the fountain (which is the most obvious structure today besides the amphitheater itself), but this did not fundamentally change the nature of the area. The final form (with Temple of Venus and Rome, Baths, fountain, adjoining the Forum to the west) does not really suggest a place where you would go to buy fish sauce or a new pair of shoes. It was monumentalized, probably busy with foot traffic, but not cramped in the manner of a bazaar. Shopping was probably further south towards the porta Capena, or else in the massive shopping district constructed by Trajan for this purpose on the opposite end of the Forum.