Answer
Yes — create a custom Event Action of type 'Streams Message' configured for the standard system event BACKGROUND_JOB_IS_PROCESSED, with whatever arguments are needed, and add the target group of users (rather than just the job's creator) to the action's 'To' section. Enabling this event action sends the Streams message to that whole set of users whenever a background job is processed, rather than being limited to notifying only the user who originally created the job.