HEllo
I would check you /etc/syslog.conf
You should check where your auth.log records records are stored. Typically its in /var/log
The only thing I can recommend then is to set up a cron job to parse the file and look for occurrences of sshd etc.....
(you should see entries when you do cat /var/log/auth.log | grep ssh ) You can further refine your grep and stick it in a bash script to output to a file, cat /var/log/auth.log| grep ssh > my_ssh_log.log
That done you need to then email it to your self. This would require you to (IMO) use the SMTP server on your ubuntu server, and given your slightly concerned about hacks or access , opening up and using port 25 for mail is a bad idea. That means you would have to integrate to an external mail server, which again I would not recommend.
I would instead recommend you use something like rsync to automatically copy across the latest file produced and store it in some directory on your machine.
Hope it was some small bit useful