If I share a site the email is not being sent. This function does work with everything else like if you share a document or a list. This is happening for all users and all sites. The relays on the email servers are setup correctly.
This old question might be the same problem/solution. The Share with email does not use the farm settings as it sends as the currently logged in user, e.g. Joe has shared a site with you. You might need to open up your relay(if using exchange) to allow this send as behavior.