Your monitoring solution sends its notifications via the very same internet connection.
How will you ever know that your sites are down? If you have the resources you can set up monitoring from an external source as well, but what if that's not an option? If this scenario keeps you awake at night, then perhaps you should look into Monitor.us. Monitor.us allows you to monitor not only HTTP and HTTPS, but other services such as FTP, SSH and MySQL. The service is easy to setup and the free version offers many options. Here is the dialog that you get when creating a new monitor:
Monitoring intervals for the free version are 30 minutes, but more frequent checks as well as increased monitoring sources are available for a charge. With the free version you also get a weekly email report giving you a summary of uptime and SLA's.
I use Monitor.us in conjunction with Opsview so that if my ISP circuit goes down I still have a failsafe. Between the two, it makes a pretty complete monitoring setup.