Should one of them be closed?
If so, which one?
Does the choice depend on where there have been answers?
Should authors of answers copy (or perhaps even copy-then-delete) these?
If one should be closed, what close reason do you consider appropriate?
The latter might depend on which of them you close, since I can come up with an easy off-topic reason for closing on MO, but can't think of anything which would be on topic for MO but off topic for MSE, except perhaps for some work-life questions.
Yes, one should be closed as off-topic, and migrated to the other site, where it can be closed as a duplicate. It's a bit painful that we can't close as a duplicate of a question on another site, but that's the present state of things.
I think the majority of cross-posted questions will be off-topic for MO and on-topic on MSE, but not all. I see you're not a MO user, but it's clear that there are many questions which will receive better answers if asked on MO than on MSE. This doesn't count as "off-topic" for MSE, of course. It's up to MSE how they'd like to handle these questions; migrating to MO will sometimes be helpful to the asker.
The suggestion in this post is only about cross-posts, by which I mean that a user posts the same question at the same time on both sites.1 But very probably this is the same thing what the OP meant by cross-post.
If the questions is left on both sites and it is, for example, closed on MO and answered on MSE, MO will be left with the closed question. If the question is migrated to MSE, then it will be closed there and this time MSE will be left with a duplicate question. Both scenarios increase noise on the site (in the first case on MO, in the second case on MSE). Moreover, in the case of migration we are adding to the workload on moderators.2
Suggestion: If a users makes two posts with exactly the same question almost simultaneously on both MO and MSE and this question is undoubtedly better suited for MSE, the following steps seem to be reasonable (in my opinion):
- Close the question on MO, leave a comment for the OP explaining this question is better suited for MSE. (And perhaps also a comment explaining that it is bad manners to post the same question simultaneously on two sites, especially without mentioning this fact.)
- Left the question here for some time; sufficient for the OP to see the comments.
- After some time (maybe 2-3 days) the question can be deleted from MO. (Perhaps again the first user casting the vote to delete could leave the comment that it is deleted because there exists MSE duplicate and the question is not suitable for MO anyway.)
I think this approach might help to decrease number of unnecessary duplicated questions and perhaps it would make the work of mods a little easier.
My suggestion is only for questions having no answers. (A question having upvoted or accepted answers cannot be deleted anyway.) And, as I have already said, it is not about questions where there was sufficient time between posting it on MSE and on MO.
1 I do think that in some instances there might be legitimate reasons why essentially the same question might be posted on MO and on MSE. For example, one of possible reasons is different level of expected answers. Usually greater mathematical maturity is expected at MO and the typical answer at MO might be too terse for some of the askers. But in any case, if someone decides to ask the same questions at the other site, they should explain their reasons for this and link to the other question. This problem has been discussed elsewhere and this is not what I wanted to address by this point. I have included this footnote mainly to stress the asymmetry between MO and MSE; my proposal above is only about deleting the duplicates on MO. I am not sure whether the same argument would work as a reason for deleting a question on MSE.
2Unless the migration is done by regular users - if enough of them vote to close the post and choose the option to migrate to Mathematics. In such case, it should be checked whether the migration is not going to be rejected because of tag-incompatibility - details are explained here: How to (re)tag such that migration works?