I managed to configure my site to receive e-mails to default e-mail from the users e-mails. Then cron creates nodes and nodes' authorships are correctly attributed to the respective senders (users). The only problem is that nevertheless they are authors of the nodes, for some reason they can not edit or delete their own nodes. On node/$nid/edit or node/$nid/delete Access denied message is given. I suppose mailsave is missing some kind of hook to not only attribute nodes to respective users, but also to enable them to actually manage their nodes.