I was under the impression that if a user was tagged in the Newsfeed i.e. @carl that SharePoint 2013 would send a email to Carl telling him he was mentioned in the Newsfeed. I cant figure out where this functionality lives in any setting. Can you help?
I cannot comment to clarify but I assume you have it connected to Exchange and other emails are being delivered e.g. when you reply to someone's comment or when you subscribe to changes in a file that is then changed?
Check your Exchange for security and delivering of anonymous messages - adding sharepoint's address to Exchange so it's not being discarded as spam.
What do you have set as in Central Administration -> System Settings -> Configure Outgoing Email Settings? Did you allow Exchange to relay messages from that address?
Or you can install SMTP server on your machine and have it send that way https://technet.microsoft.com/en-au/library/cc263462.aspx