My daughter has, at varying times been: a cat, not real because someone else is dreaming her, a fairy who created the universe and rebirths itself every million years because it gets bored otherwise, a cat but one that can talk, a dead drunk starfish, me, her mother, and sometimes even herself.

She's also remembered little details that stuck out to her (sometimes verified in photographs or by returning to a place) but which everyone else missed.

The null hypothesis, therefore, is that children are fantastically perceptive and unexpectedly creative (including with identity), and this makes the information transfer potential far more fraught than you might expect (especially for a rare phenomenon).

There's also extensive work on memory implantation which shows it's rather likely, so you have to be extremely careful that the experimenter also has no information whatsoever about the supposedly reincarnated person when talking to the child (or that whoever is taking down the accounts has no idea of who the person is).