Of course, because they are EU citizens. All EU citizens can travel freely. But immigrants need a visa since Cyprus is not in Schengen. This is the case with all non-Schengen EU countries, like, immigrants living in London need a visa to go to Paris, but a Romanian doesn't need visa to go to London. An immigrant in a Schengen country can travel to any other Schengen country plus new non-Schengen entrants: Cyprus, Romania, Bulgaria and Croatia. For example, i am an immigrant in Lithuania (Schengen) and i travelled to Cyprus and Romania with no visa (i can't even technically apply for one), plus many Schengen states.

This situation is not uncommon. [Reunion Island][0] is a French province, but it is not in Schengen neither. Any European can go to live and work to Reunion Island, but a foreigner with a Schengen Permit cannot.

[0]: https://goo.gl/maps/TAJoEYLp7m32

Immigrants in the UK (right to remain, not just permanent resident) can travel freely in the EU (and CTA without even a passport).

Ah, I see, thanks. So immigrants aren't yet full citizens. In retrospect, that makes sense.