We didn't need to work together for space progress in the Cold War. Just the opposite, in fact, the competition drove space progress then.
The problem is funding and short sighted politicians. NASA has given some of, if not the greatest results of any government agency, and yet it ends up on the chopping block and not, say, the Department of Defense (horribly bloated, by the way). They're politicians. They got where they have by winning popularity contests, not by being smart, so I'm unsurprised that so few of them are able to see the blatantly obvious value of a strong space program now that the United State's doesn't have any other nations' to freak out about.
That's why I wish China the best of luck with it's space program. The EU, as a conglomerate of electoral democracies whom are mostly official US allies, just isn't threatening enough to put fire under a politician's feet. Japan, India, and Russia aren't really active enough in the field, and while the US has been politically rather cold towards the last two it isn't comparable to the Cold War itself. China, on the other hand, is still being used as a stick and carrot by politicians despite how unsustainable Chinese politics actually are. So that's why "Chinese Space Supremacy" is one buzzword I wouldn't be complaining about being used in politics.