I think this is "The Giant's Tooth" by Bruce Coville. It is collected in Odder than Ever.
I read it in an old copy of Realms of Fantasy and I found a list of stories from that period here. Summary, such as it is:
Edgar Twonky didn't intend to get eaten by a giant while he was heading for Cottleson Fair. Sometimes these things just happen. What was even more startling was what he discovered once he was in the giant's mouth . . .
I remember the story a bit. I recall him saving several people and they eventually string a rope across the mouth to help save more. I remember them carving out cavities (much to the giant's dismay) to make homes. In the end, he falls out of the giant, then starts making plans on how to get back into the mouth.
Found this preview at smashwords. Definitely the story I remembered.
The bit about the older man he fails to catch is this:
While he continued to work on his new dwelling, Meagan taught him how to snatch things from the tide of food and rubble that poured down the giant's throat three times a day. When she "went fishing," as she called it, she first secured a safety rope to one of the chairs inside the tooth. Normally she pulled things in with the help of a long pole that had a hook on one end. But if something particularly good came rushing past that was too far out on the tongue for her to snag simply by leaning for it, she would fling her whole body onto the surface of the tongue, then use the rope to haul herself back.
Once they saw an old man go past, but he was all the way in the center of the mouth, and they were not able to reach him despite their best efforts. His cry of despair as he disappeared down the giant's gullet echoed in Edgar's dreams for many nights afterward.