Some of the disagreements in earlier answers seem to stem from different understandings of peer review. From the definition stated in the question, it seems totally reasonable to think of disagreements in the mishna and gemarah, or in the rishonim, achronim, and later works, as constituting peer review. Those who disagree are typically "people with similar competences", and are, in some sense, "evaluating" the work of others. Functionally, machloksim sometimes motivate different sides to refine their original opinions, or even occasionally retract them. In this sense, I think it's reasonable to answer "yes" to the question.
But for those who are familiar with peer review from their experience in the world of scholarly research and publication, the definition is likely unsatisfactory, and these people will probably be inclined to say, no, there is no real and effective peer review in typical rabbinic literature, not in the talmud, nor later.
In conventional scholarly communities, when someone has a good idea they try to write it down as best as they can, clearly stating their main idea, bringing as much possible evidence for it, hoping to convince other scholars that their ideas are original, interesting, and compelling. They then send off their manuscript to a journal, and editors of the journal send the manuscript to other experts in the relevant particular field to evaluate. If someone does potentially ground-breaking work in treating chronic myeloid leukaemia, for example, the editor won't send their work to just any general doctor, or even to a general oncologist, or even to a general research oncologist, but instead will send it to someone in a very specialized subfield who is qualified to evaluate that particular idea. Multiple such referees will review the paper and provide recommendations to the editor (who is usually not a specialist in this particular area) about the quality of the submission. Maybe the idea is terrible, or maybe it's a brilliant, but has already been developed by others. Or maybe it's original, interesting, and overall compelling, but can be improved in various ways. The peer-review process functions to reduce the number of bad papers published, and to improve the quality of the ones that are published.
In this sense, of course rabbinic literature is not peer-reviewed. Sure, R' Meir and R' Yehuda, the Rambam and the Raavad, or the Ktzose and Nesivose might regularly disagree, but that's not what members of functioning scholarly communities normally think of as peer-review. And yes, journals like Hakirah, Tradition, The Journal of Halacha and Contemporary Society, The Torah Umadda Journal, etc. use some kind of peer-review, but neither the talmud, nor the geonim, rishonim, achronim, nor most contemporary lamdanim and poskim, do anything remotely similar. Instead, the vast bulk of what's published is self-published (i.e., if I can raise enough money to print my chiddushim on X, then some seforim press will take my money and publish what I wrote, irrespective of whether it's excellent or terrible), or else is published by kollel's, yeshivas, etc, who typically publish almost everything submitted. Like your high-school student newspaper.
So, the answer to the question then, I think, depends on what is meant by peer review. If all is meant is that we allow for some room for disagreement, then sure, it's peer reviewed! But if by the term we have in mind a particular rigorous process that helps ensure (but of course doesn't guarantee!) the quality of what's published, then I think the answer is unfortunately no.
