This is an important issue to get right.
The two math sites are clearly different and are aimed quite
deliberately at different but overlapping populations of users,
questions and answers. The same question, were it to be posted on
both sites, would receive very different kinds of answers and
levels of attention at the two sites. It seems to me that we should develop a
guiding policy for migration to help each question find the
home that is most suitable for it.
As an active user of both sites, I have often found questions on
math.SE that would in my opinion receive much better attention at
MO (and conversely, of course, but that is not the issue I am
discussing). Interesting but technical questions are often found here
at math.SE that end up with comparatively low views and slip
unanswered off the main page, whereas on MO these questions may
find vigorous interest and activity. Even when a question is answered here, it might have gotten more illuminating or otherwise interesting answers on MO. Although it is obvious that
the math.SE user base includes numerous mathematical experts in
diverse mathematical topics, nevertheless the user populations do
differ in aggregate and the questions I have in mind have not
found the same success here (in terms of views, attentions,
answers and votes) that I expect they would find on MO. This is a
(Some examples, found very quickly, might include: Forcing questions, Generic reals in forcing iterations, Preorder of forcing notions, a question on $\Delta$-system,
Comparing countable models of ZFC; surely others can find many more examples. I believe that each of these questions should be migrated.)
Thus, I think we should develop a guiding policy for thinking about which questions belong on which site.
To help with this, let me begin by disagreeing with Henning's
assertion that migration should occur only at the request of the
OP. It is important for us to retain community control over this
issue, and the OP's opinion is often not the most relevant. Some
users do not realize that their question is much harder than they
expected, or out of modesty or anxiety or what have you, may post
their questions here on math.SE, but they really should be on MO.
At the very least, we should amend Henning's suggestion to also
give credence to the users who answered the question, since
these users also have an interest in placing their work where it
can be appreciated best.
But even more, I would say generally that we should give credence
to the community views about where such a question may belong. We
should encourage people to make comments suggesting a migration
where it is suitable, and simply allow the voting mechanism for
migration to occur as a natural process on the site.
The policy I propose is in a sense already the default policy of the site, since it is the policy that is actually implemented in the software, which allows users to vote to migrate questions. Henning's policy, in contrast, would seem to require either a software change (which seems unlikely) or else acrimonious exchanges when people vote to migrate without an OP, or with an absent OP.
Surely the boundary between the sites will evolve organically, and our ability to migrate questions will allow us to refine it.
I propose that we think initially of the dividing line as something like the
dividing line between undergraduate mathematics and graduate
mathematics, which I take to be similar to the line between
mathematics-as-interesting-subject and mathematics-as-profession.