The intra-EU VAT special scheme OSS is a significant simplification to intra-EU cross-border sales and anyone applicable should register. Immediately.
Both EU and non-EU companies can register for OSS. As the OSS only handles intra-EU cross-border sales, the goods have to reside in the EU. For the non-EU companies this means they have to have a VAT registration in an EU country in order to register for OSS.
EU registered companies require a VAT registration in the country of incorporation to be applicable for OSS.
OSS registration is very simple, it really is a tick-a-box in the national tax administration portal.