Chance Encounters is the third of the three story types in Onimusha: Way of the Sword, alongside the main Way of the Sword story and Mysteries of Kyoto. Even after part of Edo-Era Kyoto has been cleared of corruption, its residents may still need help, and Chance Encounters are those requests.
Optional, Not Mandatory
As with Mysteries of Kyoto, only a few Chance Encounters affect the main narrative, and none are required to finish the game. Completing them earns various rewards, following the same clue-icon system used for Mysteries of Kyoto: people holding a Chance Encounter clue are marked with an icon above their head, and speaking to them starts the encounter.
Rescuing Townspeople
A related field activity plays out directly in the streets rather than through a marked NPC: Genma attack townspeople out in the open, and stepping in to defeat the attacking Genma before it is too late earns a token of the rescued person's thanks. This can happen while simply passing through an area, without needing to seek out a specific Chance Encounter first.
Achievements
Steam's achievement list ties two achievements directly to this rescue activity: one for rescuing 30 townspeople in total, and a separate one specifically for rescuing a merchant.
How Chance Encounters Differ from Mysteries of Kyoto
Both optional story types share the same clue-icon system and the same rule that only a handful of entries touch the main narrative, but they are tied to different circumstances. Mysteries of Kyoto arise directly from advancing the main Way of the Sword story and a certain individual's requests, while Chance Encounters surface specifically in areas that have already been freed from concentrated Malice, once the immediate danger to a district has passed and its people can be helped with smaller, more personal problems.
Practical Notes
Because rescues can happen opportunistically while exploring, it is worth staying alert to Genma attacking bystanders even when not actively pursuing a Chance Encounter lead; the 30-rescue achievement adds up faster when every attack on a townsperson is answered rather than ignored.