Should there be certain types of questions migrated to Matter Modeling? (e.g., that seem to be a better fit.. like ones specific to a particular program)
Yes. Judging from my personal experience, there are quite a few questions that remain unrecognised here, or even badly received, that may well be a good fit there.
I usually go about and ask in their chat about these cases and post comments here. (I am doing that for other sites, too.) Occasionally I just send it over. I'm a member there, I have a bit of experience migrating now, oftentimes I can make the right call.
Apart from that, everybody can comment on these questions and raise custom flags.
Should there be a "community specific" close to migrate to Matter Modeling?
No. These are still pretty few occasions which can easily be handled manually. We don't need to crowd our already crowded queue. Most of these cases would probably have to be expedited anyway.
Apart from that, I don't want to deal with the formal change request.
Furthermore, if we open up this line, we need to discuss other sites, too. And from my point of view there is no evidence for having these either.
How would this be made clear to new users, who may not know of one or the other site, or why some questions may be better suited for one site or another?
Following the above outlined workflow, you'll get a sense of the user and how they will further interact with our site or our partners.
In most cases where I asked about migration, the users were very forthcoming and thankful. It's easy to miss our partner site as they are still quite new.
It's probably for this reason alone that I would want to handle these cases manually. It gives the whole thing a more personal touch.
How do we minimize people posting questions in both places?
If you follow the above and adding a note that cross-posting is discouraged, you've probably done as much as you can do.
My general experience is that cross-posting isn't a big problem for our site. Nevertheless it should still be discouraged.
I'm generally adding a note on both (all) posts with links to the other site(s). I'll then usually close the post at our end.
I'll tell them to delete the other posts and custom flag for this one to be reopened, or do the same for the other site.
If it's on matter modelling, there's a good chance it fits well there and poorly here, so it remains open there and closed here. If it doesn't get deleted by the cleanup procedures, this could be expedited, too.