You could honor a 'content="nopin"'. It's be interesting if that became a standard (similar to nofollow or robots.txt).
[1] https://help.pinterest.com/entries/21063792-How-to-prevent-p...
If someone has a copyright complaint with a public Pinry it will need to be directed at that individual because we really have no control over it and even if we did honor something like a nopin tag it's open source, it'll just get removed if someone wants to post copyrighted material.
And if not, what do you think would be a typical usecase, i.e. who would want to run Pinry-powered website?
But this is the same question you should have for a Blog I'd think, if you host copyrighted content on a blog then you'll have the same issues if it's a public blog.
If you are permitted to download, you are permitted to move the dowoaded file around for yourself.
Doesn't Google/Bing do this with their image search?
In practice, I suspect most copyright holders will send a DMCA takedown anyway, either to you or to the company hosting the site, if you're on a VPS or PaaS service.
Pinterest can get around this by claiming that it was the user who pinned the photo, not them, and they also allow websites to opt-out from pinning (and some websites like National Geographic actually do that). But if you self-host, you are responsible for what is posted...
I am asking because, if I could legally run a website where I would publish smaller version of any photo I find on the Internet, then well, there's a great monetization potential there. But I don't think it would be so easy.