When changing a Page in New UI (Experience Manager or XPM), the page is checked-out. What you might be seeing for other users is expected behavior--other users should not be able to edit the page in the CME or within XPM.
Also, you should be restricted from editing content page for even the same user that has a different session (e.g. viewing the page from another browser).
When editing the page with the same user and session, you should be able to add multiple components. The page is checked out. Editing content on the page should be "editing components," rather than the page itself.
Let us know if you're seeing something else.