Reviewing is completely voluntary in all senses. I don't know the percentage of research active mathematicians who have ever reviewed for MathReviews or Zentralblatt, but it is certainly bounded away from 100%. It is relatively rare for a research mathematician to write reviews in any quantity over a period of more than a few years: I think it's something that many of us try out for a while but don't stick with in the long run. For a personal touch, I am a 2003 PhD, I have written about five math reviews, and I have gotten overdue notices on about four more. I would rather write nice, insightful reviews for half of the papers that I get sent, but I haven't yet figured out what do about that.
I agree with Tim Porter that, as a graduate student, you should discuss this with your advisor. I think that if any of my students asked me about it, I would mildly discourage them from writing reviews, for the following reasons:
(i) It is not closely enough related to your thesis work to be a good use of a mid-to-late career grad student's time.
(ii) Prospective employers are generally not going to be more excited about your application because of the reviews you have written.
(iii) Most graduate students -- even ones who have already done some research work of their own -- don't have a broad enough perspective to write insightful reviews. (Or, more positively put, they will have a broader perspective later on and will probably write only a bounded number of reviews in their life no matter when they start.)
Finally, I should say that refereeing papers is a totally different story. I feel that it is an ethical (and karmic) responsibility to referee at least as many papers as one submits. A graduate student can make a good referee for some papers, because they are less likely to feel that it is beneath them to read the paper line by line and really make sure it is correct.
There are three separate issues here.
1) How to clarify whether the proof is correct? You should start with making a serious good will effort to understand what is written (which amounts to redoing all the bad notation, splitting things into small steps, etc. to the best of your abilities). If this fails, you should state as clearly as you can what exactly the problem with the argument is and hope that some expert will figure out who is right. Of course, first you should send the full account of your effort to the author reproducing all the parts of the proof you understand and showing clearly where you are stuck and why. Just to say "your notation is bad here so..." won't accomplish anything: at best, he'll make a local correction that will move the real issue somewhere else and you can play this shifting game forever.
If he still fails to address the issue after that, request the help of some third party
sending the same account of your effort together with the paper. Again, it is important that you demonstrate your good will and decent understanding of what is written in the paper before you raise your objection. Without this, you just won't be taken seriously. Make sure that you understand everything that precedes the unclear/incorrect step and that you make it clear to everyone whom you want to ask that you understand it. Nobody pays attention to people coming out of nowhere with zero credentials and doubtful qualifications. If your first words are "I don't understand ... and I think it is wrong", the most likely answer will be "Go learn ... ". However if your first words are "This argument starts with using ... to establish ...", your general credibility goes up immediately (provided that what you are saying makes sense). The more times the person agrees with you on the issue before you raise the question, the more likely he is to take you and your objection seriously.
It is your moral duty to make a real effort trying to understand the proof before making
any public comment on its correctness but it is also your moral duty to report a problem
with a proof when you are convinced that you see one. Note that it is completely normal
in mathematics to make bad mistakes occasionally and it is completely normal to fail
to understand correct arguments now and then. The priority/reputation chase has distorted the general attitudes beyond recognition, of course, but the heart of the matter is still the search for the knowledge, not building/preserving/destroying reputations and relationships.
Even if you are wrong on the account that the proof has a gap, you may be right
on the account that it is unclear (assuming that you have a decent education in the subject, the fact that you fail to understand the argument is a clear indication that the paper is written not in an ideal way). So, the clarification may help innumerable poor souls (like graduate students) who may have the same difficulty but just do not dare to ask questions. You risk to look like a fool, of course, but the only way to avoid looking like a fool occasionally is to always be one.
2) How to avoid the confrontation? At some point there may be no way and all that you'll be able to write to the author something like "It seems very difficult for us to understand each other. Since the issue is principal, the best we can do is to seek an opinion of a third party. I'd appreciate your suggestions of whom we should ask. I'm thinking of (put the list of experts you know)". This may not save your good relationships but, at least, will clear you from any "doing things behind the curtains" charges. After that, send your doubts to both the people on your list and the people on his list, if he provides one. If he doesn't, it is his problem. There are three possibilities: 1) you'll be backed by some expert, which will make the author harder to ignore you; 2) someone will explain to you why the proof is correct, and 3) everyone will ignore your request. In that last case you may have to seek the opinion of general public but not before you double check your argumentation.
3) Is MO an appropriate place for this discussion? It is not what it was intended for but if you finally decide to seek the general public opinion and post your objection on arXiv or somewhere else (in the way I outlined above; let me emphasize once more that unless you are Terry Tao or Tim Gowers you should demonstrate both good understanding of the matter and your good will before anyone will bother to take a look at your objection in honest) I see nothing wrong with making a short post containing the corresponding link in this thread.
In brief, if you really want to figure things out, I would advise that:
a) you make a good effort putting all your thoughts together in written. Create a clear "case" starting from the beginning where you agree with A on everything and talk in the same terms and stopping where you have an objection.
b) present this full writeup to A and wait for his comment.
c) if it doesn't result in anything meaningful, present this writeup to a few experts or,
at least, people whom you feel to be more knowledgeable in the subject than yourself.
d) if you are totally ignored, ask yourself why that may be the case and, if you see nothing wrong with your written argumentation (if you see nothing wrong with what you keep in your mind, there is a good chance that you are just blind), present it to the general public.
I don't think asking a graduate student of A is a good idea. First, many graduate students are totally incompetent in anything beyond their thesis project and in such cases, you can just as well use a parrot for communication. Second, if the student is actually good and you are right, you'll put the student in the position where he will have to tell his adviser that the paper is wrong. This doesn't go well with many people.
Best Answer
Of course, reporting an error benefits the author, so I advise to write to him/her. A polite form is: Dear X, I am reading your interesting paper Y, and I have difficulty in understanding why A implies B on p. C. I will appreciate it very much if you explain this to me (help me to understand).
I received such letters several times. In most cases I was able to explain. In one case I published a correction. And of course I am thankful to all who wrote these letters.
When I was a student I wrote to one famous mathematician: Dear X. Let me bring to your attention that conjecture W which you state in your paper Y seems to be proved by Picard in 1880 in his paper Z. Do you think his proof is correct?