I'm more of a mind that it -should- mean -demanding- residents/fellows assume more responsibility as their program director feels they become more qualified to do so. If you're willing to send a resident/fellow off to be an attending, you should be willing to let them sign out essentially independently by the time they complete the program.
Unfortunately, billing &/or insurance seems to pretty much prevent even senior residents from becoming "autonomous," except in limited circumstances -- some programs allow senior residents to independently handle certain frozens/overnight transplant biopsies/autopsies/common not-usually-tumor specimens such as gallbladder or appendix. However, IMO, the expectation should be that by (if not before) a resident completes the program their attendings should be quickly reviewing & co-signing rather than sitting down for a joint sign-out, except occasional cases the resident flags (i.e., knowing their own limitations). If the program attendings aren't confident in this, then they probably shouldn't be confident graduating that resident to go work somewhere. But my impression is that a lot of programs only pay lip service to the concept of graduated responsibility and leave it at teaching junior residents and handling tumor boards basically on their own (work most attendings consider "busy work"). Or they only progress to this level with rare "special" residents, instead of truly expecting it from everyone.
So while I agree that Gene_'s examples describe graduated responsibilities, personally I think the end point should be an expected/demanded one functionally equivalent with that of so-called junior faculty. But for the most part I'm not sure I see that happening with regularity outside of the fellowship setting.