Follow up question: After buying a cut of meat, how would the customer take it home? Was it wrapped in paper or cloth, carried in a bucket, etc. Did they have something resembling butcher paper back then?
I wrote a tangentially related answer about the different types of Ancient Roman pigs and what they looked like here: https://www.reddit.com/r/AskHistorians/comments/56343k/in_the_first_season_of_rome_theres_a_scene_where/
If you haven't seen it, look at the picture of the pig preserved after the eruption of Mount Vesuvius.
I have not seen the HBO series so I have no idea what happened in the scene. But the series ends somewhere around 30 BC and at that time it would be very rare for any common people to eat pork for any reason other than a religious ceremony. I think it's safe to assume that more than 95% of the population, at around the end of the Republic, did not eat pork even once a month. Or if they did it was in tiny portions bought from a food stall, not at a butcher. It was too expensive, so it would be unthinkable for anyone but the rich to ever buy a leg or side of pork from a butcher.
Over time it would become more common, although fish remained more popular in the south of Italy. Under the rule of Aurelian (270-275 AD) pork was given away with the dole, although it was probably in small portions or happened infrequently.
Columella and Cato the Elder wrote about how to raise pigs, prepare them for slaughter, slaughter them, and preserve them. But as far as I know they didn't write about the profession of butcher. So I don't know what butchers in the city of Rome did exactly. But any meat that was fresh would be local and probably extra expensive.
The majority of pork was butchered away from the City on farms and preserved before being carted off for sale in cities. I don't know what the average farm would be like, but the archaeological remains of a pig farm were found at Settefinestre that had a complex of 27 pig sties.
Preparation Methods:
Two popular methods were dry curing with salt and smoking. Adding brine was unpopular, but some pork was preserved by layering it in salt in large jars where it would create its own brine and then flipping the jars every few days. "Apicius tells us that sides of pork or beef or tenderloins of both are pickled in mustard, vinegar, salt, and honey, completely covered. He also says that fresh or cooked meats suspended in a vessel and covered with honey will keep in winter, but only a few days in summer." But note that honey was very expensive.
The making of sausages "was already well known by the Babylonians and Chaldaeans and was certainly used by the Greeks long before the Romans."
There are plenty of places to find recipes and preparation methods if you want to recreate some history. Here's one for the preparation of Roman salt pork: https://qcurtius.com/2015/06/19/the-roman-preparation-of-salt-pork/
Now I'm hungry and have to find some food :/