How does a test like the Natera Panorama distinguish the child's genome from the mother?
As you correctly discerned, distinguishing the fetal cell free DNA from the mothers cell free DNA in the blood stream is one of the major challenges in this type of diagnostics.

Consequently, the results that can be directly linked to the paternal DNA are the most reliable.

In the case of the example you give, the high accuracy stems from the fact that in case of Monosomy X, there is no indication of neither a paternal Y nor a paternal X chromosome. While the Y chromosome is the most easily discerned, a paternal X chromosome could also be detected because in that case the cell free DNA would contain fragments of three distinct X chromosomes.

This publication summarizes the issue nicely:


  Once the cell-free DNA fragments have been purified, small differences
  between the fetal and maternal DNA sequences are exploited in order to
  make a specific fetal diagnosis. To date, the majority of studies have
  focused on the detection of paternally inherited sequences that are
  entirely absent from the maternal genotype, such as those on the Y
  chromosome of male fetuses. This target is particularly attractive as it
  comprises a large portion of DNA which is not otherwise present in 
  women.
  
  Variable regions of repeated DNA (short tandem repeats or
  STRs) can be used to identify paternally inherited sequences, by comparing
  the number of repeats present with the paternal and maternal sequences.
  Paternal alleles on the autosomal chromosomes can also be detected, if
  they are known to be absent in the maternal genome, but this requires
  detailed sequence knowledge of the paternal genotype of interest, as
  well as detection methods that can distinguish DNA sequences that
  might only differ by a single nucleotide. Importantly, all these methods
  rely upon the fetus inheriting a uniquely paternal sequence that is conveniently
  located for a particular diagnosis.


The reliability of these tests can apparently be improved by enriching the fetal cell free DNA:


  Selective enrichment of fetal DNA, based on a difference in the
  average physical length of fetal and maternal DNA fragments, which
  can be exploited to increase the relative amount of cffDNA. Fetally
  derived DNA fragments are generally smaller than those that are
  maternally derived, being predominantly ,313 base-pairs in length
  (Chan et al., 2004). Therefore, by using standard size fractionation
  to select only DNA fragments ,300 base-pairs, circulating cffDNA
  can be enriched such that it comprises around 70% of the total cellfree
  DNA (Li et al., 2004b), prior to detection and identification by
  either PCR (Li et al., 2005) or mass spectrometry (Li et al., 2006).