I wasn't familiar with the term "intersubjective," so I looked it up. From the wiki article, it says: ""Intersubjectivity" has been used in social science to refer to agreement. There is "intersubjectivity" between people if they agree on a given set of meanings or a definition of the situation. Similarly, Thomas Scheff defines "intersubjectivity" as "the sharing of subjective states by two or more individuals."
That sounds to me like it's just a subjective opinion that lots of people agree with. That doesn't change the fact that it's subjective. There are lots of people who think Star Trek is fantastic, judging from the groups I am in on Facebook. That would, according to your implied argument, elevate the "Star Trek is fantastic" idea above a simple subjective opinion, yes?
And I think that a person's moral position IS up to the individual. If God came and said to you that you could commit murder and you would get no punishment, not in this life, nor in the afterlife, would you do it? I get the feeling most people would say no. But why would we still conclude that murder is wrong even if God told us it would be okay? The only explanation is that we get our morality from ourselves, and we would refuse to murder because we believed it is wrong.