IMO the correct way is to include subset of JRE that's actually required for given application. This can lead to relatively small total application size and deployments that do not require any knowledge of java and java world from end user. Having one JRE for all apps on the system is nice but it's usually not needed.
Why "installing" almost anything requires "access rights" is still confusing to me. I don't understand why there's not been a movement to 'install' things in to my own home directory structured, vs "c:\program files" on windows "/usr/bin" and such on linux. - Fall back to c:\mystuff or ~/bin.
There are downsides to this sort of duplication (like security bug fixes not being automatically applied to all apps using a certain JRE revision) but in real-world practice, the benefits outweigh the downsides.
Windows' SxS DLL system has the same basic issues but it solves so many distribution, UX and versioning problems that the downsides are totally worth it.