This seems to explain the technique they're using to study this:

Ancestral recombination graph (ARG) inference (2) is a powerful starting point for such an analysis. An ARG can be conceptualized as a series of trees, mapped to individual sites, over phased haplotypes (chromosomes) in a panel of genomes. Ancestral recombination events, or sites at which chromosome segments with different histories were joined together by historical recombination, form boundaries between trees. Each ancestral recombination event manifests as a clade of haplotypes, all of which descend from the first ancestral haplotype to have it, moving from one position in the tree upstream of the event to a new position in the downstream tree (3). ARGs are complete descriptions of phylogenomic datasets and present for recombining genomes what single trees present for nonrecombining genomes, i.e., a complete description of their genetic relationships.

We run [our ARG] SARGE on a panel of 279 modern human genomes, two high-coverage Neanderthal genomes, and one high-coverage Denisovan genome. Using the resulting ARG, we map Neanderthal and Denisovan ancestry, ILS, and the absence of both across modern human genomes.

We find that approximately 7% of the human autosomal genome is human-unique and free of both admixture and ILS. Roughly 50% of the human genome contains regions where *one or more humans* has archaic ancestry obtained through admixture. If deserts are further restricted to regions that contain a high-frequency, human-specific derived allele, i.e., a substitution that can be assigned to the human lineage (hereafter “human-specific regions”), then these comprise only 1.5% of the assayed genome.

It sounds like a fancy graph-based Venn diagram of the genome overlap between different groups comprising sets of individual genomes. I'm still not 100% clear on their definition of "human-unique" - i.e. are these regions where all 279 sample human genomes are identical, or maybe all 279 had one or more modern versions of the allele vs. any of the samples still having an ancestral version?

They go into detail on some specific genes:

Several patterns emerge when considering genes with high-scoring human-specific mutations, and we highlight some of these key findings. Gene ontology terms related to mRNA splicing, processing, and export are enriched in genes with high-scoring mutations (table S7). Of these, we find a regulatory mutation affecting one—LUC7L3—that is somewhat tissue specific (tau, a measure of tissue specificity scaled from 0 to 1 = 0.713), most highly expressed in cerebellar tissue, and annotated to be involved in splice site selection. Its paralog LUC7L and the gene KHDC4, both of which also have high-scoring (top 50th percentile) regulatory mutations, are involved in the same process. In addition, the gene NOVA1, which harbors a nonsynonymous mutation in the top 95th percentile of our score distribution (Fig. 5C), is a neuronal splicing factor that regulates splicing of genes involved in synapse formation within the brain (43).

Which makes it sound like maybe yes, all modern humans have the "new and (hopefully) improved" version of these genes while no ancestral samples did.

https://advances.sciencemag.org/content/7/29/eabc0776