Should authors be required to report a paper is being resubmitted? This is probably more relevant to conference submissions than to journal submissions, but still. 

Some CS conferences have recently started to employ a resubmission policy: if your paper got rejected from a prior (relevant) conference, and you are resubmitting,  you have the option of attaching your previous attempt’s reviews and a cover letter explaining how you addressed them. 

You don’t have to do this. There are very good reasons to believe that doing so is beneficial. For the authors, it offers a chance to show improvement and a willingness to improve (thus deflecting potential repeats of prior criticism). For the community, it prevents p-hacking of the review process (resubmitting until you luck out and get your paper accepted). 

What are the potential downsides of making this process mandatory?
The obvious downside is that you can't verify it. You could have a policy which says "if your paper has been rejected elsewhere, you must attach your previous review and a cover letter explaining how you addressed the issues". However if the authors don't do it anyway, how would you find out? The only chance is to invite either the editor or reviewer(s) who reviewed the previous submission. The odds of this is small, especially since you don't know who edited/reviewed for the other journals, or even which journal it was.

The other danger is that you open a potential path for fraud. Currently the most common form of fraud is for authors to suggest as reviewers people who are linked to them (or to themselves using obscure email addresses). With this policy, one could easily write the "previous review" oneself to give the paper a sheen of reputability. The new conference cannot verify these previous reviews are legitimate. In principle this should not fool the editor, but in practice it is all too easy. Example of the former ploy fooling journals; this is likely to manage as well (at least some of the time).