The rate of VAT on services supplied to non-business customers in other EU countries normally depends on where the supplier is located or, for services related to land, where the land is located.
A UK-based business supplying services to individuals in other EU countries will currently charge UK VAT. However from 1 January 2021 the UK will not be treated as a member of the EU, so services supplied from a UK business to individuals in EU countries will not be subject to UK VAT. Many professional and legal services, including advertising and data processing, will be treated as outside the scope of VAT so no VAT will be applied.
Services supplied between businesses across international borders have been treated as outside the scope of VAT for many years but the customer usually has to prove they are a business by supplying their VAT number.
Your EU suppliers may wish to charge VAT to your business from 2021 as they may not accept your UK-issued VAT number as valid. That is incorrect. Your UK VAT number will remain acceptable proof of your business status from 1 January 2021 and beyond.