Investigating how Romanians often think and talk about the Roma people in their country may be more harmful to their reputation than thinking the beggars are non-Roma Romanians.
Things are more complicated than that. Romania is home to more than one Roma people. The Roma who go to Finland for organized begging are almost exclusively from the south of the country. They can actually seem quite foreign to Romanians from e.g. parts of Transylvania. There, the local Roma are often associated with different ways of making a living than begging, and speak a different set of languages preferentially. Some ethnic Romanians from Transylvania may be very tolerant about their local Roma community, but feel that those particular Roma people from particular counties are ruining things for everyone. Even Transylvanian Roma people can feel that way about the Roma from elsewhere.
I don't think it's fair to single us out, though. This social problem, integration, is present in almost exactly the same way in Hungary, Slovakia, Bulgaria, Serbia. To a lesser degree it's also present in Spain from what I know.
It's far from a simple problem. As the Finns are discovering.
Finland is home to a centuries-old Roma community that arrived via Sweden. They too are seen as abusing social services and not integrating, the same general stereotypes as in southern Europe. (Members of this community can often be identified in Helsinki from their distinctive clothing, the women wear traditional skirts.)
So, Finns are already aware that Roma populations often live in tension with the major ethnicity of a country. However, the confusion about the ethnic makeup of Romania along with other Balkan countries persists due to the coincidentally similar names for these ethnicities, and lack of interest among the Finnish population about educating themselves about a region that seems far away and to which very few go to on holiday.
