One major issue is that, because of this, they will sometimes have a vested interest (conscious or not) in blocking the publication of your paper if it shows flaws in their own.
Having a good editor who can discern the difference between a reviewer's relevant responses and 'turf warfare' is probably best one can wish for.
Of course, take my semi-anonymous random internet opinion with a grain of salt (I've dropped out of college not once, but _twice_, so what do I know?). However, I'm moderately good at computer science topics, and have spent the last year reading and implementing papers in detail in pursuit of a fairly niche topic, and, what I was dismayed to find, is that most papers are absolute, complete and utter garbage. Like, honesty embarrassing. And that's before you even get into the fact that _you can't reproduce their results_.
Encountering this has been something of a crisis in my entire belief system, to be honest. Although, maybe I've just been exposed to what "real" science looks like: a big, gross, often wrong mess of assumptions about the world that takes a long, long time to correct.
I would agree that some papers are terrible though, I've seen a few I would never have let through review including in the very top journals.
I like the Plos mega journal approach but we can't do away with peer review altogether otherwise what separates science from the conspiracy posts that circulate on Facebook ...
So often as a reviewer I have avoided asking for citation of my own work, even if highly relevant, as I perceive that to be an abuse of power.
I've also started a comedic collection of terrible reviews I've received, though it took me 15 years to gain the confidence to know when these are an indication of the reviewer's failings, not mine.
That's noble, but isn't it counter-productive? If you're reviewing a paper whose readers would ultimately benefit from a citation of your work, then would the paper not be improved by citing it?
I think there is some aspect here of just finding reasons to dislike the reviewer (though I've certainly had reviewers where those reasons were pretty easy to find).