The quote has nothing to do with who is trained in signal processing in the abstract, and everything to do with who has a detailed understanding of this particular machine.
page 3:
"The only persons who have the right critical mindset to assess this paper are in the LIGO Scientific Collaboration"
I believe instead it is simply saying:
"The paper has no obvious flaws that anyone unfamiliar with LIGO could point out. Therefore the best people who could assess it are the LIGO team"
What's the point of peer review, then? Why bother publishing detailed experimental reports and providing access to data if the team gives it the two thumbs up and no one on the team sees any flaws. Their word should be good enough.
The fact that one team is more qualified to analyse a paper that another team has produced has nothing to do with whether or not that paper should be independently reviewed.
The claim is that only the people working in the project are qualified to criticize; that's almost the opposite. People who are working on the project are more often than not the LEAST qualified to criticize, because they're heads down in the weeds and have too much invested in the particular details and experimental choices that were made.
Moreover, for LIGO specifically, not every scientist is motivated to work for a high-risk, long-time-till-result endeavor. In fact, I'd suspect the people most likely to approach things with the right critical mindset are antiselected from working on the project.