Answer
There's no native, narrowly-scoped permission for this — the platform doesn't expose "see everyone's background jobs" as its own grantable privilege separate from broader administrative access. A workaround some have used is building an IAL (or equivalent read-access object) against the underlying background-job tables, exposing it through a Quick Report, and granting just that report to a limited permission set — but treat this as a real security decision, not just a configuration step: it exposes details of other users' background jobs (and whatever data those jobs touch) to whoever gets that permission set, so only do this with a clear, deliberate justification for that specific visibility rather than as a routine convenience grant.