Yes, we are going to need to establish some international law about exploiting resources located off the planet. The concept of who "owns" an asteroid is not easily decided. Is it the country of origin of the astronomer who first discovered it, or the science team that first remotely determined it had unusual density, or the country that first sent a spacecraft to fly by at close distance, or the country that first lands on it, or the country that first returns a scientific sample, or does it remain available for any country to take as much as they can manage to take back to Earth? And, who is going to enforce that law, and how?The rumors are that the calculated density (and that is very hard to figure out exactly) is very close to the density of Gold. It this thing turns out to be an actual huge chunk of Gold, I fear a huge race to grab as much of this (or remotely claim it as theirs) as possible. I fear a huge human folly powered by greed. Even if it isn't Gold, it could have other very valuable metals, again triggering greed. Well, we'll all know in time. Oh, BTW. The thing is 500,000,000 miles away. Not very close at all, between Mars, and Jupiter.
To me, it is not rational for one country to be able to lay claim to a whole asteroid or moon or planet or star. At least, not to one that it cannot actually move as a whole object to bring it to Earth or a mining base that is off-Earth.
But, from a financial perspective, if it is only economical to develop and execute a mining operation when most of the material gets extracted in order to make a profit, having 2 competing entities working to acquire the same asteroid's resources could doom both to financial insolvency.
Actual cooperation is what is needed. With enough cooperation, laws really aren't needed. But, our species seems to have a short supply of cooperative urges.