A while ago (about 2009 I think), when I was a freshly baked Juniorprofessor, I was joking with a colleague that it felt a bit awkward to still be part of studiVZ (a social network for students that was very populuar in Germany at that time – I think actually a Facebook clone). Being a bit smug about our new positions we thought that there should be something like profVZ for guys like us.
Not precisely in the same direction but somehow related: I heard in this post be Peter that the AMS is working on some online community for mathematicians. Go and read the post in which Peter argues why this is a great idea.
By the way: The German pendant of the AMS, the DMV, already has a default social network for all its members (I blogged on this in an earlier post). This is pretty hard to find on the DMV homepage and it seems that almost nobody is using it for anything (except the few people who contribute to the forum). If I see correctly, the DMV social network is using the JomSocial software, a community tool for Joomla! sites. It looks like JomSocial has quite some flexibility and a lot of tools available. However, I have no idea if this could be adapted in a reasonable way to the needs of a mathematical social network (whatever they are, precisely). The AMS has produced great tools already (just think of looking up papers and references in MathSciNet) and I expect that a AMS social network can be great too. However, I would prefer if such an initiative would be run by a worldwide organization, i.e. the IMU, basically just to emphasize that the social network is intended for the whole mathematics community. Anyway: I think that the mathematical community can greatly benefit from a professional social network and that this could be much more useful than the other general ones (just as MathSciNet is in general much more useful for a mathematician than Google Scholar is).