I am a PhD student and I have been contacted today by some other PhD student with a very odd request. He is from a different university in a different country and I don't know him at all. He is asking me to review his own paper before submitting it to some journal. Is this a common thing to ask? What is the best response in this case?
If he's a complete stranger, I can understand the situation feeling a little creepy to you.
I suggest you ask him how he chose you for this role. Ask him where he's studying and who his advisor is. Look up the advisor online to see if everything looks above-board.
His response to a short email with a couple of neutrally posed questions may help you figure out if you are comfortable with his initiative.
Since you don't know him at all, it can be a little weird, but the ice can be broken if you exchange some e-mails. As others have said, you can be more comfortable with him if you talk more. Also, you can ask if you two can collaborate on a project. You didn't give any details on the particular paper, if it seems a good idea, and also his publication record. If the paper seems good(even if it is not perfectly written yet), IMHO you can start a collaboration.
From a social and scientific standpoint, you should agree if you have time and interest in the paper. You may get a contact who works on similar themes and you may collaborate in the future. As others have noted, the person will also likely be on the hook for reading one of your future manuscripts as well.
There is, however, another perspective, which is that of someone who is concerned about intellectual property. Many large companies have strict "no unsolicited ideas"-policies, because they do not want to risk future disputes. Something similar may apply to you: Say you are already working on a new method for solving problem X, that you have not currently published. You recieve a paper from a stranger who has had the same idea. Suddently, by accepting the manuscript, you may in the future be accused on plagiarism of the same idea. The reasonable thing would perhaps be to collaborate on a joint paper since you have the same idea, but people in the academic sector are not always reasonable.
This depends highly on the nature of your field of research. In my experience, these types of issues are likely more prevailent for research with commercial applications. Universities are increasingly becoming concerned about intellectual property rights as well.
Barring the less-likely (and nefarious) reasons (he wants to make contact with you for some other reason, he is spamming all phds at your university) such request is perfectly legitimate.
You did not say if the paper is from your area of research, or whether he refers to some of your published work - that may be important in determining the reason why he is asking you for review.
Actually, checking the quality of one's paper with your peers (other phd students) is a very common advice given to phd students by their supervisors (if they are any good). True, this mainly refers to the other people in a lab, but there is no reason that people would not go beyond this.
What does he gain by you doing the review? Well, if you will criticize his paper, he can fix the problems and has better chance to be accepted first time around or with minor revision. The reasoning is, even if you are only phd student, if you can find problems with his paper, the more experienced reviewers will find them too (and probably some additional problems you will probably miss). It is not called "peer review" for nothing (check the definition of the "peer" in the dictionary).
What do you gain by honoring his request (although, I should stress, you are in no way obliged to do the review, especially if it is far from your own area of research!)? Well, first, you will get insight into what your peers (your competition perhaps) are doing. You may get a invaluable chance to peek into the good paper which will be officially published in 6 months or even longer (although, until it is published, you will be bound by the confidentiality). Yes, you may find out that the paper is worthless and that it was all waste of time, but hey, welcome to academia and the world of peer reviews. It will not be your last time.
So, I would suggest, that you ask him why he contacted specifically you. If he is following your publications, then perhaps you will be interested in what he is doing. If you find out that his research is unrelated, politely decline and say that you are not expert in his area.