In principle, my services are subject to Austrian VAT (Mehrwertsteuer) at 20%. I charge this tax to all clients in Austria.
When the client is a business in another EU country, the reverse-charge procedure applies. This means that I do not charge VAT. To document the status of the client as a business, I must put their VAT ID number on my invoice.
When the client is a private person in another EU country, I must charge 20% VAT.
If you are paying for my service out of your own pocket, you can probably claim the expense as a tax deductible. In Austria, depending on your top tax band, you should be able to get up to 50% of the price back. Ask your tax adviser.
Similar arrangements may also exist in other countries.