This might sound controversial to some but my understanding of it all is that the whole society concept is a bit outdated. I have a feeling that most of those organizations or societies have older roots and try to live on the reputation of what it once was. In my own field I don't know a single person who's a member of the national/regional/global organization. I know some who used to be, though...
The only benefits I have figured out so far are discounts on journals, conferences and in some specific cases travel grants for younger scientists. The "problem" here is that the first two never really mattered much, in my experience. I don't know how it is at other institutes but we normally get access to a wide variety of journals via the university library network (in other words, I never had any reason to subscribe to journals) and likewise we are required to apply to travel grants in order to go to conferences, so having some percent discount does not make a huge difference (we either get the money to go to the conference, or we don't).
To refer to some of the points mentioned:
as I mentioned in a comment to Shion's answer, providing access to relevant literature should be a responsibility of the department/university/research institute. It is such a fundamental part of the work so that not having access to literature would be a service-professional not getting a hammer/screwdriver from his employer. Access to literature is as much of a vital tool for us as a chisel might be for a carpenter..
some of the other benefits, such as representation towards policy makers, could be extremely region-dependent. For instance; as a European resident, I see absolutely no reason to be a member of ACS, which is one of the biggest societies relevant to my line of work.
organizations/societies do organize conferences but you still have to pay to attend, plus I have yet to come across a society-exclusive conference so far. Note that I do not dismiss the possibility, but rather stress how uncommon it is.
All that said, I do not advocate that societies are useless. I only suggest that given the meager salaries we get as grad students, I think you can come up with better use of the money that would have been spent on the membership fee.