It's not one or the other, it's both. They both do original reporting on their own and act as a loose network of smaller investigative journalism organizations. It truly depends on the task at hand, but it's usually very useful to get some of the local investigative journalists involved, as they're the ones that both understand the language and are able to put the leaked data into context. Usually ICIJ is the one that publishes the English version of the story and their local partner(s) publish the same thing in the local language.
Actually, I'd say there's three such networks: ICIJ, GIJN and OCCRP. They're not really competitors, each serving slightly different purposes, and there's plenty of collaboration and overlap between their members.
Source: I'm not in that world anymore, but I knew about Panama Papers long before it was public and have my name in the credits in some of the collaborations with ICIJ.
Actually, I'd say there's three such networks: ICIJ, GIJN and OCCRP. They're not really competitors, each serving slightly different purposes, and there's plenty of collaboration and overlap between their members.
Source: I'm not in that world anymore, but I knew about Panama Papers long before it was public and have my name in the credits in some of the collaborations with ICIJ.