These quests are typically chosen at town quest boards. Side quests are optional opportunities for rewards that are independent of the party’s guild quest. Each side quest has its own requirements for completion. The party completes its side quest at the end of an Encounter Phase in which that quest’s requirements are met (unless the quest specifies a different timing).
Though individual adventurers may choose which side quests to gain, active side quests are shared between the entire party. Your party can have up to 4 active side quests at a time. If the party would gain a side quest and exceed this limit, it must either discard 1 of its current side quests or the side quest it would gain. The party can also choose to discard 1 or more side quests at any time.
There is no inherent penalty for leaving an active side quest uncompleted unless that quest’s card says otherwise. If you cannot meet the requirements to complete a side quest, you cannot complete that quest but may keep the card until the end of the session.
Side quests are not specific to a single province. Instead, they often refer to map features, such as landmarks, that are found across Tamriel.