Informed traders do know the cross section of the privately known signal between each other
Yes, that assumption just means that every informed agent knows the true distributions where signals come from (i.e. roughly speaking, they know the parameters that define them).
The terminology &quot;cross-sectional distribution&quot; comes from the usual interpretation that instead of having literally 2 agents, in reality, you have a continuum of agents that come in 2 types. If that is the case, due to the law of large numbers, the empirical distribution of signals at any point in time will be equal to the theoretical distribution of signals. Therefore, knowing the cross-sectional distribution is the same as knowing the theoretical distribution. Since these distributions are normal, the latter corresponds exactly with knowing the mean and variance of signals of the other (type of) agent.