Make sure the semi gloss is latex paint. I suspect it isnt, since latex paint doesnt go over popcorn very well. Since its painted already, you can use whatever kind of paint you want, but, I would put a coat of primer on it, just in case its oil based paint, so you dont end up with a peeling ceiling. Most pop corn ceilings are not removed because they are dangerous, they are removed because people want flat ceilings. If you dont mind the popcorn ceilings, then keep them, and paint them, and stop worrying. You may not even have asbestos in your popcorn. Not all popcorn ceilings did. In any case, you are not going to have asbestos floating around.

This may b e of help: https://www.google.com/search?q=how+to+remove+popcorn+ceiling+that+has+been+painted&client=safari&source=hp&ei=gQTvYZyeFvyxqtsPhY6X8Aw&iflsig=ALs-wAMAAAAAYe8SkWpUIAg77fE-3vbweKNB-qHVG1s_&oq=how+to+remove+popcorn+ceiling+that+has+been+painted&gs_lcp=Cgdnd3Mtd2l6EAEYADIFCAAQgAQyBQgAEIAEMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeMgYIABAWEB4yBggAEBYQHjoECAAQQzoFCAAQkQI6DgguEIAEELEDEMcBEKMCOggIABCABBCxAzoLCAAQgAQQsQMQgwE6CAguEIAEELEDOggILhCxAxCDAToLCC4QgAQQsQMQgwE6CAgAELEDEIMBOgUILhCABDoFCAAQhgNQAFjOT2CRd2gAcAB4AIABWogB6RiSAQI1MZgBAKABAQ&sclient=gws-wiz

skim coat over it (encapsulate it). Bunches of videos on youtube showing you how.

Looks like sand was added to paint or some type of ready made brush on sand texture product was applied. Probably no concern about asbestos but have it checked for your peace of mind. I would investigate sanding the ceiling vs overlaying with more sheetrock. Power drywall pole sanders that hook up to a vacuum are quite inexpensive. South of $200 inexpensive. You can rent a pro-level sander, too. But I would just buy an inexpensive one outright. Sell it when you're finished and recover a good chunk of what you paid for it.