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This is the response I received from Omnigraffle Support on 13 September:

If you copy Object A, then click on the 2nd canvas in the Canvases sidebar, then paste, the object will paste in the same location. If you've clicked elsewhere in the canvas itself before you paste, it will paste where you've clicked.

There's also a newly-discovered bug wherein if the canvas you're pasting to is not set to the same zoom factor, this doesn't work properly.

[Suggestion to use shared layers]
I didn't know about the "click where you want to paste" feature before, and it is occasionally to blame for unexpected Paste in Place behaviour.

More often, however, it's a bug. And it doesn't only happen on canvases of different zoom levels, as I can prove with a document I currently have open. At any rate, I don't know how "newly discovered" the bug they're referring to is, as I first reported it on 2008-05-30, again on 2009-04-03, and again on 2010-04-30. And again in this thread.