As an outsider looking in on the Scientific process, I am not really sure how applicable my opinions are, but I see these as useful changes.
Basically, in reverse order, my suggestions for science to adopt are as follows:
Papers in databases need to have fields related to reproduction studies, and it needs to start becoming a prideful part of the scientific process; just as there is a lot of pride and money, researchers should start to thump their chest based on the reproducibility of their work, actively seeking out contemporaries and requesting a reproduction study as part of the pubilshing process, and subsequently updating.
The papers published themselves should take a moment (perhaps no more than a paragraph) to include a "for media" section that outlines the "do's and don't's" on reporting on the research. For example, cancer research should clearly state examples of acceptable understandings in lay person terms as a sort of catch for sloppy reporting. Something like "Do not write "cure for cancer found" or "Effective treatment", instead write "progress made, etc". Basically put a sucker punch to outlandish headlines and reporting right in the paper itself, and let journalists who want to be sensationalist embarrass themselves.
This seems like two very simple acts that could raise the bar for science a bit.
Of course, they probably do know this and just choose to ignore it because "Unverified Study that MIGHT Point to M&M's Being Good For You" won't get as many clicks as "M&M's Are Good For You Says New Study!"
it's not so much checking for the public purpose, it's for others.