I think they'll collaborate. But it will look different to what you've seen in space so far.
The space companies will probably need to protect some patents and sustain their business models in a way you would not see if it was only agencies working on the projects. So I see collaboration taking form in that agencies, whose job it is mainly to enable space-based science and push technologies enabling space exploration, will look to join complementary private projects. Sometimes as the main customer, sometimes as a research partner.

I can imagine many of the current jobs done by agencies will be taken over by companies (providing launch capabilities and infrastructures) allowing agencies to focus more specifically on their own tasks related to science and cutting-edge technology developments.