Qiqi's Talisman will heal characters attacking the target to which the talisman is applied as long as the damage dealt to the target is a) based on the attacking character's stats and b) dealt by a character which is currently on-field.
Healing from Fortune-Preserving Talisman (Qiqi Q) has cooldown (approximately 1 second). In-game tooltip only mention the target taking damage as condition. However the talisman has cooldown on how fast it can heal.
The healing from talisman only occurs every ~1 seconds and healing ticks will not increase against multiple targets (suggesting cooldown is shared between targets).
To add to this, the healing cooldown appears to be shared across the team.