The transacrion is constructed by a coordinator and signed by each participant.

This is a simple example of a two person coin join. You can see how even if you know who owns the inputs you would not be able to tell who owns the outputs since they all look the same.

There would be no way of knowing from just looking at the on chain transaction that the first four outputs belong to Alice and the last two belong to Bob

Perhaps this example is also helpful.

Imagen you know who owns each input on the left. How would you tell who owns the output on the right since they all loom identical?