Theoretically, then, if you’re not an EU resident, mobile companies aren’t obligated to offer you “roam like at home” pricing. That’s potentially bad news for visitors from elsewhere in the world, who plan to buy a SIM card in the first EU country they visit, then use it across the continent.
Practically, though, many carriers are unlikely to bother trying to restrict their service like this. I spoke to a customer service representative for EE, the largest mobile company in the UK, who confirmed international visitors will be treated the same as any other prepaid customer. As long as they abide by the standard fair use policy, they can roam across the EU like anybody else.
My advice for international tourists planning to use a SIM in this way, though, is to ask this specific question before purchase. Each operator is different, and especially in cheaper Eurozone countries, some may look to keep their costs down by enforcing this part of the regulations.
We mainly wanted something for prepaid data versus for voice and text. 
