Given that pushbullet only accepts signups via Google or Facebook accounts, what is the recommended way of sending alerts from (for example) "My site Service "?
I tried connecting an additional address to my account an this also presented me with a Google / Facebook login option.
Is the only way to send notifications from "My Site Service" to create a Google or Facebook account for "My Site Service " ?
You can create a google account for your service, I have some test accounts I created that way.
One way to get the behavior you want though might be to create an OAuth Client https://docs.pushbullet.com/#oauth which can send messages on behalf of whatever user grants the client access. This is for example what IFTTT does, and the messages show up as if they are sent from IFTTT, not the current user.