Depends. If you talk about rights as given by some law, then the law subjects are the ones who bear rights. In some cases the law subjects are individuals, in other cases (international law) these are explicitly not individuals.
Nations do not have "rights to vote", for instance: it all comes down to the individuals that make up those nations.
It's really hard for me to understand how groups of people would have rights, if someone could elaborate on that, that would be much appreciated.
Nations do have the right to vote, just look at the UN General Assembly (well, not counting Taiwan or the Sahrawis).
Regarding rights of individuals: If country A commits a violation of international law, and an individual B who is national of country C is affected by that, then B cannot sue A directly. Instead, C must bring the case against A before an appropriate court like the ICJ. This happened e.g. in the LaGrand case[1] between Germany and the US.
On the other hand, if A finds that C owes it money, and C doesn't pay, then any assets of B that happen to be under the jurisdiction of A can be confiscated to pay for the debt. This happened e.g. in the Icesave dispute[2] between Iceland and the UK.