All EU based customers are subject to VAT. However, a VAT registered EU entity can make purchases from VAT registered suppliers in different EU countries without VAT being paid, but the transaction is not VAT free.
Suppliers outside EU, should charge VAT to EU based customers unless exemptions apply.
Under old rules, VAT liability was determined by place of supply, now it is based on point of delivery.
This is not an expert view - ask an accountant for exact details.