One effect of the place of supply rules is that UK suppliers can find themselves liable to register for VAT in other EC Member States. Liability to register (and any consequent requirement to appoint a local tax representative) is governed by the rules as they are applied in the Member State concerned. You should not offer any opinion as to whether a UK business might be required to register for VAT elsewhere within the EC. Enquirers should be advised to contact the appropriate VAT authority. Contact details can be found on the website referred to at VATPOSG1100.
Overseas businesses with no place of business in the UK may nevertheless be making supplies in the UK and so required to register here. Further details about this can be found in Notice 700/1 (Should I be registered for VAT?).