To reject cosmopolitan morality means to absolutely reject thinking about what is best (or worst, or the lesser of evils) at the scale of the world, the universe, the people, the masses, the class, the nation, or anything claiming to be everything. Instead, choices may be made based on what is best for yourself--and, optionally but in most cases, what is also best for those who you are in a substantial relationship with (not just in a milieu, social scene, so-called community, organization, or Facebook group with) who are on your side of conflicts.
In other words, thinking through ethics on an intimate, small scale. A refusal of thinking about what are the problems of the world, or humanity, and what to do about them.
A further implication is one I find particularly interesting. Because we are indoctrinated into believing we are part of a people, race, nation, world, etc, for most of us this small-scale thinking will require us to actively train our disbelief in the existence of society as such. To believe that there are 7 billion people existing on the planet is to subscribe to a faith concocted by politicians and preached by their agents. A suspension of disbelief that can be an interesting diversion from real life, but one that I would rather leave to the sphere of fancy and games.
Without this faith, and without throwing individuals into crowds and masses where their individuality is drowned, how many can I say exist? Things start to feel crowded when I try to think of more than 7 at a time.
A rejection of legal morality implies that--in addition to rejecting the authority of some governing power claiming to represent millions of people who I don't even believe exist to know and dictate what people should do to best secure a goodness for an everyone I reject--I don't even think that rules are good at securing goodness in my life and small circle.
Things at this small scale are more unclear and fluid than they are in the imaginary interactions of the machine-like masses, and I think this is good. I can't come up with any hard and fast rules for what is always good... (the closest thing might be, paradoxically, that no rules are always good)... and I don't particularly care to. What is more challenging and rewarding is subverting the existing rules. Sometimes this will mean refusing the idea of good and evil, the assumption of any straightforward link between action and consequence, the belief in the ethical subject, and so on.
As far as I can tell for most people who use the terms, a personal morality or ethics just means that one differs from the dominant universal moralities, and/or doesn't accept those as actually being universal or having authority. Each individual has their own principles they stand by, which they use to determine what is right and wrong on the scale of their own life.
My problem with this is its tendency to simply fall into the democratizing sphere of moral relativism. This way of thinking has been rising in dominance vs. universal morality for some time, and at least one factor that should lead anarchists to be critical of it is the place it has had in government since at least the French Revolution if not earlier.
I am running out of words here where it gets most interesting, but will mention two texts which take the discussion much farther than my current energy level is able to. The first: Tiqqun's "Introduction to Civil War" which describes Empire as a neutralizing force of liberal social control which distances us from our own power and undermines the intensities of the ethical conflicts between forms of life. The second: "Its Core is the Negation" by Alejandro de Acosta which gestures to an ethics emerging from countless refusals. http://theanarchistlibrary.org/library/alejandro-de-acosta-its-core-is-the-negation