How do you earn opportunites to review journals or conference papers? Critical reading is a very useful skill for most PhD students (and postdocs and researchers in general). 
Instead of assuming that everything you read in a scientific paper is right, it's useful to learn how to evaluate the paper critically: e.g., to question its content, to identify shortcomings and limitations and ways it could be improved.

Experience with reviewing papers is a powerful way to gain experience at critical reading of papers. Writing reviews for journals and/or conference helps a PhD student learn to get better at reading a paper with a critical perspective.

Unfortunately, review opportunities for PhD students are rare.

For instance, it is rare for PhD students to be invited to serve on program committees or asked to review papers. In addition, some may think that PhD students are not competent enough to write correct reviews of scientific papers.

How does one solve this problem?  How can a PhD student get opportunities to practice reviewing papers?  What would you suggest to a PhD student who wants to do some reviews?
I'm going to address your question, but first, I have some issues with it generally:


  I think that one of the problems of most PhD students (and postDocs and researchers in general) nowadays is that they don't read scientific papers with a critical judgement. They often think that everything is right in a scientific paper; they're not used to doubting its content...


This is manifestly not true in my opinion. Indeed in my experience (and I've seen this shared by others), I've watched faculty members reign in students who had torn into a published paper for what were essentially minor methodological flaws that wouldn't change the substantive findings of the paper one way or the other. I think a far more common problem is "failing to see the forest from the trees".


  But, anyway, review opportunities for PhD students are not many.


They can be. I've reviewed 4 or 5 papers for journals in my time as a PhD student, and a disheartening number of conference abstracts.


  How does one solve this problem?


There are three ways I've gotten papers to review:


Your advisor puts in a good word for you. Essentially, a journal asks them to review a paper (or if they're an editor somewhere, a paper hits their pile) and they redirect it to you, either formally or informally.
Publish. All of the papers I've reviewed are in areas where I already have a well received publication, which bypasses the "Journals don't think PhD students are competent" problem.
Some conferences put out calls for reviewers. Keep an eye out and sign up.



  What would you suggest a PhD student who wants to do some review?


Publish. The strongest way I've ever ended up getting papers to review has been from papers I've published. Talk to your advisor. Look out for opportunities - I've seen at least three calls for reviewers in my time expressly open to students. This also gives you an experience being reviewed, which is important both for honing your own skills as a reviewer, and something you need to learn how to deal with.


  Are there any open-to-review journals where one could train oneself?


You don't need a journal to do this. One of the most useful things you can do to train is to get a faculty member to support a journal club where, in addition to presenting the paper, the student writes a critique in the style of a review. Not only does this force you to read the paper you're presenting more closely, but it will let you learn in a protected, mentored environment rather than "out in the wild".