I want to find partners to study some post-graduate topics together online (because I'm pretty much out of steam as far as self-study goes, and I have problems finding a decent grad school). Are there good web sites that facilitate this kind of partnership?
I'm pretty sure there is no site like that, although it sounds like a good idea. Actually I sort of use this site for that purpose, and am in a similar position to you.
One nice thing to do is just read every question to get idea's for new topics to study, and then do some quick research to get up to speed so that you can at least understand the answers if not come up with them on your own.
Another nice thing about this site is that when I am working on some idea or other, and get stuck, I will post a question to try and generate some outside input. I find this to be extremely useful most of the time, and there are a lot of people on this site who are more than generous with their time and energy.
I think a good philosophy to have about these things is the following: If a good intentioned person who is genuinely working hard to try and understand something asks a question that I can answer, then I will take as much time to explain it to them as it takes, because that small effort on my part will save this person a large effort on their part. On the other hand, if I detect intellectual laziness in one form or another, I don't feel the same motivation to take the time to explain what I know. Therefore, if you want partners to help you self-study because you "ran out of steam", then you might not have a lot of success finding people to work with, instead you should work on getting that steam back, and when people realize you are working hard and dedicated to self study, they will be more than generous in response to your questions.
We could make a website. Here is some suggested functionality:
If someone were enrolled in a university course, this might make an excellent opportunity to lead an online version at the same time. Further, students might be able to use a course as the backbone for independent study (I know I took lots of these). This would serve to unite traditional and non-traditional students of mathematics, which in my opinion may be the most useful function.