For example, see https://www.elsevier.com/about/policies/copyright (under "Author Rights") for what Elsevier permits you to do with your own work.
On the other hand, publishers have sometimes filed lawsuits against sites where authors share their papers, e.g. ResearchGate: https://www.nature.com/articles/d41586-018-06945-6
Elsevier have also sent takedown notices to universities where academics have made the final version available on their institutional websites:
https://www.washingtonpost.com/news/the-switch/wp/2013/12/19...
https://osc.universityofcalifornia.edu/2013/12/elsevier-take...
https://news.harvard.edu/gazette/story/newsplus/elsevier-tak...
https://blogs.library.duke.edu/scholcomm/2014/01/28/setting-...
Use https://v2.sherpa.ac.uk/romeo/ to check. Also, EU projects in most cases now require open publishing and publishers make exceptions even when OA is forbidden (“self archiving is allowed if mandated by the funding agency”).
Always seemed like a grey area to me. We didn’t really distribute the copy of the paper with the journal/conference’s name + copyright - though a perhaps a line under the title: “To be published in…”
For the final, not open access published papers they certainly do.