Under what circumstances is it acceptable to post the same question on Math Stack Exchange and on Math Overflow? More specifically, I want to know if it is acceptable to duplicate the question if I have gotten no answers in here and some days have already passed. I understand that some questions might require more than a few days to be answered, but in the case of, say, a reference request I see no point in waiting.
Sometimes you want to have pizza for dinner. And that's fine. You call the local pizza place, and you order pizza which is delivered to your house usually within 30 minutes.
What you don't do is neither call several pizza places and see which one gets first to your place; or call a falafel place, and order a large pizza with extra mushrooms.
When do you call a second pizza place? When you have waited 45 minutes and the delivery guy didn't show up with your pizza; or if you have given up on the pizza you might order a falafel instead.
What does this analogy teach us?
Asking a question online is like ordering a pizza. You ask on one place, asking at several places at once serves little more purpose than calling all five local pizzerias and ordering the same pizza, just because you're very hungry.
Sometimes, as Andres Caicedo remarks, your question is a pizza but for that purpose MathOverflow only sells falafel, so it has neither pizza nor mushrooms, to help you. Therefore depending on your question, it might not be suitable at all for MathOverflow.
If you are certain that the question is good for both sites, pick one, ask there, and let it sit for a while. What's a while? Say a week. If you placed a bounty at some point, let it sit through the bounty period and see what's going on. If all that failed to get you closer to an answer, either delete the question and post it on MathOverflow, or raise a flag and ask for it to be migrated there.
Be prepared for the possibility that your question is not welcomed there, and it will be downvoted, closed or migrated back to MSE. It may happen. Don't take it personally, you called to a vegetarian pizza place, and asked for sausage on your pizza, and they declined your order.
If you are not certain, you can always post a question on meta.MO and ask there. Explain the situation and ask whether or not this question is likely to be accepted well. If the response is positive, go ahead and repost/migrate. If not, try posting a bounty on your question here, try talking to people on the chat system. But remember not to overdo it, there's a fine line between asking people to help you, and harassing them with borderline spam.
That been said, of course there are possible merits for having two questions on two sites. But those should be aimed for receiving different answers. On MSE you may ask for a particular explanation for a particular solution, whereas on MO you might ask for a more general outline of different solutions (or references to them, and read yourself).
But in that case, don't forget to post cross-links in your questions, and be very clear that there are two questions and that they have different aim.
In my opinion, it's not a good idea to compare questions at MSE/MO with pizza/falafel orders. The qualitative difference is that when you "call the local pizza place" you have immediate feedback, and you can specify details: you usually ask people there, if they indeed can make what you want (sometimes this is impossible, say because they do not work today), how long will it take them to do this, etc. It is very important that
they give you a promise: "OK, no problem, wait us for 30 minutes", or "...for 1 hour", etc.
If you imagine that when posting a question at MSE you immediately receive a feedback like "OK, this looks easy, wait for a day", or "2 days", or even "a week" -- then there would not be a problem at all, I am sure almost all crossposts would disappear. Similarly, the problem disappears if you receive a negative feedback, like "you know, this seems to be too difficult, you should ask this at MO".
But this never happens. The main problem is uncertainty: usually you know nothing about whether somebody thinks about what you ask, or there are no such people at all. And very often you can't estimate the difficulties: whether your question is difficult or not - is not clear if you are not a specialist in this field. Using Asaf's comparison,
you call a local pizza place, where nobody speaks to you, where you don't have a possibility to ask whether they can make what you need, or not, where you make order just by pushing buttons, and after that you have to wait for the response for an unknown time.
I would say, this problem with uncertainty does not exist in real pizzerias. And something indeed unexpected and serious must happen if they eventually don't bring you pizza.
I agree that hurrying in such deals looks not nice, but waiting for a week, in my opinion, is too much.
Apart from everything else there are obvious situations, where you, even being not a specialist, understand that answering must be easy, I mean reference requests about something well-known, say, about textbooks. Sometimes it is clear that all specialists in this field must know the answer -- but even in this case this comparison with pizza causes protests when you crosspost your question.
In short, I think,
the idea to wait a week should be corrected. There must be more detailed rules on when the question can be crossposted in a day, in two days, in a week, etc.
Math Overflow is for research-level mathematics (research-level means research done by professional mathematicians). If your question fits this description then you should post it at Math Overflow (because it is likely the few people here would be able to answer). If your question does not fit this description then even if you waited for 1000 years and no one answer, you still should not post at Math Overflow. Honestly, I can not see any reason to cross posts between the two sites. The only exception being migration of research questions asked at Math SE (very rare) or migration of general mathematics questions that were ask at MO (more common).
For my applied math research questions (not suitable for MO) I was able to get very good answers (sometimes) here at Math SE after posting a bounty (that was also suggested in the other answers).
I have always wondered if MSE and MO are that different when it comes to research-level questions:
As I take it a true "research" question would require research to answer. - Why would somebody do that research for me and post it online instead of publishing a paper herself?
The better term would be "research-related". I am looking for information that is already known but not commonplace enough that I would know it as well. These sort of questions arise when research touches another field of math that is not the researchers main field of expertise.
The latter sort of question is well-placed on both MSE and MO and will thus be answered on both sites.
I believe the main reason MSE exists is that there needs to be some place for rather trivial questions. People did not want to chase away competent users by prohibiting non-trivial stuff at MSE and thus MSE covers both levels while MO is reserved for the upper level.
I am not sure what kind of question could be asked on MO only, in my opinion all MO adds is some more comfort in filtering out relevant content for those interested in non-trivial issues only.
If my perspective makes sense at all, it would be reasonable to post questions with a longer "halflife" on MO. You always wondered about that particular problem, you are sure it is difficult enough to be of interest even to experts and you only require some hints to related results rather than an actual answer. You do not need results any time soon.
In such a context the stricter filtering at MO would make sense, as people are more likely to find new and interesting questions in their favourite field even after some while. It probably does not make sense if you cannot wait for a week - researchers tend to be busy...