Dwayne, thanks so much for your responses.
This issue is driving me nuts! I’m in middle of a big project, and after being careful about all of the above, I still find items connected to the wrong side.
I can have a text box in middle of the page on both sides (no place near the edges), and when I apply the master page to just one page, the text box won’t come along. When I apply the master sheet to the facing page, both text boxes appear.
If I apply 2 separate master pages to one spread, items from one will appear on the other side.

Thanks for the response. The bleed was set for all four sides – I took off the bleed on the inside.
Your advice was definitely helpful, but I’m not done yet!
Now I have the last page in the chapter, an odd page, and the master items on the even side show up, off the last page. It’s not in my way, but it is annoying. I just release the master items and delete.
But I like knowing how to doing things the right way, so if anyone can advise…