I'm in eu and don't have to pay vat for extra eu or even extra country but in eu sales.. it depends on bilateral agreement between countries i suppose..
You do not pay additional VAT, you just pay the VAT of the country your customer is resident of instead of your own country. To do that you need to be registered in all EU countries and file VAT for each one of them.
There is an alternative for online services, etc called VAT MOSS[0], mainly meant for smaller sellers which allow you to keep paying VAT in your own country without registering in all of them but you still need to apply different VAT rates depending on your customer's country. Also you need to keep records of these transactions.
Steam (and other services... well actually even Steam uses an external service for this) handle that stuff for you and you only deal with Steam.
you definitely have to add the relevant VAT, depending on your customer country, declare it in MOSS where it will be sent to every individual country.
You also have to check if your customer is any other of the countries / states that collect VAT (India, Indonesia, many US states and oh so many random countries) and make a proper declaration in each country. It's almost impossible to do it right if you don't go with a partner to handle it all.
