The mail command will use the local mail transport by default. This means the mail gets processed by procmail, which in turn will forward it to the local Mail Transport Agent (MTA) for all non-local recipients.
By default, most MTAs will attempt to deliver external mail directly via SMTP to whatever server the domain in question is using. This will most likely not work in your scenario, since your server is not a registered mail server for any domain. Also, many (if not most) ISPs will insist on you using their relay server for all outgoing mail in order to prevent abuse (spam).
If outgoing SMTP is blocked, your mail will be stuck in the local queue (run mailq to check). If not, the mail was probably rejected or summarily deleted by anti-spam measures at the receiving end.
Depending on the capabilities of the mail command on your system, you may be able to tell it to use an external SMTP server instead of the local transport.
Another (and IMHO better) option would be to configure the local MTA to use whatever SMTP relay your ISP has made available to you. This may or may not require authentication, depending on your ISP.
Your MTA is most likely Postfix or sendmail ("man sendmail" will tell you).
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