Registration for VAT in other EU countries is required when their threshold limit is reached.
The threshold limits vary in each country. When your turnover in sales to Italy within a year exceeds the threshold limit, you must register for VAT in Italy and pay VAT. In Italy, the delivery threshold limit is €35,000.
Example: An online seller is selling to Italy but is registered in Germany. As soon as he exceeds the threshold limit of €35,000 with (distance) sales from Germany to Italy, he becomes liable for VAT in Italy and has to get a VAT ID.