In general for the taxation of intra-Community distance sales you will no longer need a VAT ID in every destination country if you are already using the OSS. There are several conditions that will require you to keep or apply for a VAT ID in more EU member states even if you are registered for the OSS in your country of establishment.
For example if you are using warehouses to store your goods in EU countries you will still require to be VAT registered in those countries. Please consult GVC for an OSS tailor-made assessment for your business.