What exactly does the phrase "chimerical sharing" mean in this abstract?
This term — chimerical sharing seems to be used only in this study as a google search did not return any other results. So, it is not a standard term. I believe what they mean by this term is that the twins shared x% of the paternal genome but the sharing is not homogeneous for each cell. In other words each twin has some cells with genetic material from sperm-1 and the others with genetic material from sperm-2.

If the two sperms were genetically identical, then you would not see any chimerism. Let's now consider only one allele as a marker of paternal parentage. So, there will be two populations of cells for each allele.  If both populations are of the same size then you would have 50% chimerism in a single individual (i.e. the allele frequency). The percentage of chimerism may be different between the twins. 

Now, there may be several genomic regions at which the paternal-genomes differ. Since the paternal chromosomes would not be symmetrically partitioned between the blastomeres, the different genetic markers (alleles) are likely to have different extents of chimerism.

In this study they measured the frequencies of a few SNPs that were informative for deducing the parentage.


  Analysis of SNPs from twin amniotic fluid and parental blood showed
  that 265,489 maternal and 179,205 paternal SNPs were informative for
  parent of origin.


They noted that ~78% of the paternal SNPs were found in both the twins. 


  Of these, the twins shared 265,400 maternal and 139,155 paternal SNPs,
  which makes them 100% maternally identical and 77.7% paternally
  identical.


Note that this is just a qualitative test for the presence or absence of the alleles. This does not say anything about the extent of chimerism of different alleles. (There are some figures in the supplementary data for the frequencies of different SNP but they are not in a tabular form and hence difficult to deduce).