Yes. With a team selected, Task History and the Activity panel list every member's runs in that team, whether they were started on the dashboard or with a team project's API key. Any member can open those tasks, see their outputs and stop one that's still running.
Spending: everyone in the team also sees each member's spending.
Private work: runs in your Personal workspace aren't shown to the team, so switch to Personal with Switch Workspace in the banner (see Why is my dashboard or Task History empty?).
API and MCP: a team project's key lists the whole team's tasks too.
Deleting files: only the person who ran a task can delete its files.