I'm aware that there is no official relationship between this site and MathOverflow. However, with the moderator election underway, so far each of the 3 self-nominees has indicated that they are active both here and on MathOverflow, and I'm wondering what this means for the relationship between the communities of the two sites.
I can also provide an explanation from a personal perspective:
as Pete said, 97-99% of questions asked here don't belong on MO. I think that the ratio is actually closer to 100% than to 99%. Still, every now and then, users don't read the faq and post inappropriate questions on MO, which are usually promptly closed and the poster is redirected to math.SE. Mathematicians that are active on MO then face a minor dilemma: on the one hand we want MO to remain as useful a resource for mathematicians as it is now and don't want it to be "polluted" by questions of no interest to professional mathematicians. On the other hand, most of us see our mission in not only increasing knowledge that is available to humanity, but also in disseminating knowledge.
So, we are perfectly sympathetic with the poster who asked the question on MO and in fact glad that somebody who isn't being paid for it is interested in mathematical questions. This is what made me personally contribute to math.SE: I felt bad about telling people to "leave us alone" (which is of course not how we phrase these redirections, but some people may read them this way) and feel much better saying "let's not discuss it here, because we will disturb others, I'll see you in the other room". I imagine that similar considerations account for a large percentage of the overlap in the MO and math.SE userbase.
The designated purpose of MO is to field questions in mathematics which are of direct interest to research mathematicians (starting, approximately, at the level of an intermediate PhD student). The designated purpose of math.SE is to field questions in mathematics at all levels.
Thus the focus of MO is a subset of that of math.SE. On the other hand, if you look at the subcommunities of PhD-level mathematicians active on the two sites, the community here is close to being a subset of the corresponding community on MO.
Thus, if you have a math question that you think a PhD mathematician (often, a particular kind of PhD mathematician) will be interested in answering and someone of a lower level will probably not understand, you should probably ask it on MO rather than math.SE. Sometimes someone asks a question here on math.SE and doesn't get much of a response, and then the question gets asked again on MO. This happens with something like 1-3% of the questions asked here. Of these, I would say at least half of them are not being answered for some reason other than insufficient expertise of the math.SE readership (e.g. the question is in fact not very clear or well-defined), and then it may get rapidly closed on MO. The process of "site transfer" works best when someone who is active both here and at MO recommends that the question be transferred. This seems like an important advantage of having many dual users.
This leaves about 97-99% of the questions here which would not be asked on MO. In other words, there is actually not very much overlap between the two sites when it comes to actual questions.
What do you find specifically problematic about most or all of the moderators on math.SE being active on MO? (If it matters, none of the people who have so far stood up for the moderator position are also moderators on MO.)