I suggest not. Much as I would like and benefit from a paper sharing service, I do not want to give corporations the opportunity to aim their legal teams toward MathOverflow and claim that they were fostering copyright violation. Also, I would not want to clutter MathOverflow or meta with posts on such requests. Nor would I want a single thread to be swamped with such requests.
Something that might fly under the legal radar: personal ads. On your user page, add Wanted and Offered sections. Add a meta thread with posts arranged by general subject, and each will be a list of links to user pages. On the meta page you can add what you have to offer and what you want in general terms. For example, under the number theory post, there is a subsection on Combinatorial Number Theory. On this post I place the following lines with a link to my MathOverflow user page:
(Link Gerhard Paseman) Has: Westzynthius 1931, Wants : Jacobsthal, Kanold, any year (through 2018)
You then follow the link to see on my user page what exactly I have to offer and want, or you find my email contact and email me to request the information privately. The item above expires in 2018 or needs refreshing in 2018 to reflect my current holdings.
If you wish, I can elaborate on the details for such a system.
Gerhard "Ask Me About Sharing Systems" Paseman, 2017.07.21.