We do not have on-premise SharePoint 2013 in production but the three dev farms that I have access to all have the same odd behavior. The majority of SMTP functions work correctly (adding to groups, alerts, etc.) but the "share with" option inside of document libraries do not send an e-mail invitation. Permissions ***are*** changed correctly so you can manually send a link but this is not how we want this system to work.

There are no event log errors that I can find related to this and since other SMTP functions work, I doubt that there are firewall or mail host configuration error. They are single front-end test farms. Any suggestions on where to look?