As a general rule. In regular interactions, it's always Watcher's skill plus Party Assist. In scripted interactions you can sometimes choose a party member for the task (for example identifying an animal), and then it's that character's skill plus Party Assist. Other times everyone in the whole party needs to pass a check, and I believe PA is still active then.
There are allegedly a few cases where PA isn't active. I don't know which they are. Even when it says Watcher-only check in a scripted interaction I still seem to get PA.
For picking a lock or disarming a trap it's whoever you send at it whose skill that counts, again with PA included.
I hope I didn't say anything incorrect now.