Other experts have offered great advice here. Having a clear rubric and instructions for peer review is crucial to the process. After all, how can you expect someone to learn when they don't have any material to draw from? That being said, I've seen students, especially in middle school, rely too much on having every step laid out for them. While this may be helpful initially, it limits creativity and keeps students from taking initiative.
Mastering the writing process and improving writing skills are valuable to teach students how to be self-guided, to critique their own work, and to make sure they are communicating clearly and effectively. So, in a situation where students have no adult instruction or guidelines to follow, as you've asked about, peer review can be a very useful tool to help students improve if implemented appropriately.
I teach a creative writing class to a group of seventh graders and have come to learn that these students have had little opportunity to work on open-ended, creative assignments. They ask questions about font size, what happens if they go fifty words over the word count maximum, is it OK if they write about this or that etc. They get hung up on getting it "right" instead of experimenting and having fun in their writing. While I find myself providing more clear guidelines for them, I also want to encourage originality and creativity. This is one way that peer review can really help student writing.
From my experience, I would suggest that, to be successful, the peer review process should:
1) Include a class discussion in which students work together to formulate guidelines for editing and revision. This should be discussion of what to look for AND how to be a constructive commenter. What is particularly important here is that students should not only look for things like spelling and grammar, but for coherence, flow, and arguments/research (in persuasive writing). I find this is best done by having students formulate questions they should ask as they read. For instance: Do the paragraphs make sense in this order? Am I ever confused by what the author is saying? Does the author contradict herself anywhere?
2) Students should give positive feedback and constructive criticism. Peer editing helps students be better readers of their own work, but if they only learn how to look for what needs to be fixed, it will discourage them from reviewing work in the future. Positive feedback is not just a pat on the shoulder. It encourages students to want to continue writing and helps them understand their strengths as a writer so that they know what to continue doing.
3) Involve approximately 3 random peer editors per paper. In a class of 20 or 30 students, some will pick up on the peer review process immediately and others will struggle. If a paper is passed around to several students, chances are they will see different things. This also allows the reviewers to see what other students have commented, which will help their own review process. Most importantly, students should not be passing papers back and forth between friends.
4) Ideally, review should be anonymous. Even adults have a hard time remaining unbiased so it's a lot to ask of middle schoolers. If possible, students should not know whose papers they are reading.
Peer review, even without adult guidance, is a way for students to practice reading. It will be difficult at first, but as students receive feedback, they'll notice what is helpful to them and what isn't. Then, they can learn to implement this in their own writing. Above all, though, I believe the peer review process is an excellent way for students to learn skills beyond merely how to write well. They learn how to communicate and how to be independent learners, which is far more valuable to them than being given a list of rules to memorize.