You can request to be seated alone, but there is no guarantee you will be (unless you are concierge, I believe.) You are more likely to get your request at the later seating, though, because it is typically less crowded.
I have been on two Disney cruises so far. The first was just me and my husband. We asked for a private table and received it; we had late seating. The second cruise, we also asked for a private table, but I was less optimistic that we would get it since we were a party of three at main seating. We did manage to get our private table still, but it was placed very close to the table next to us!