Most countries have amongst themselves a double taxation agreement, where the VAT aspect is returned to the agency at the physically at the Cross-Bank-Tradezones at the border. Then remains the income without VAT, is to be levied by the local tax authorities, these regulations vary from country to country. The situation would be different, if you are living in France and work for Frenchagency, here it is important to have VAT and also charge VAT, whereas at cross borders, the invoice could be VAT-free, due to DTA (Double Taxation Agreement.
Sorry, but intra-EU VAT rules are not governed by bilateral double-taxation agreements, but are part of European legislation and, in effect, multilateral arrangements.
You may want to take a look at these threads:
With all due respect, I suggest to do some careful research before posting what could be taken as tax advice by other users.
Best regards, Ralf