Who must screen against sanctions lists?

Sanctions law applies broadly. In the US, all US persons and any transaction with a US nexus (USD clearing, US-origin goods, US person involvement). In the EU, all persons within EU territory and EU nationals worldwide. In the UK, all UK persons globally. In Australia, all persons in Australia and Australian citizens / Australian-incorporated entities anywhere in the world. Regulated financial institutions also have specific screening obligations under their AML/CTF supervisor's rules.