Do recipients of sats sent using the browser extension get automated notifications?
Any transactions through the browser extension are inherently private - so theres no public notification or automated DM sent to the user. This means the onus is on you, the sender, to let them know via a comment or message.
The reason for this is to that we keep the browser extension super simple and don't require any kind of auth to be plugged in, plus on most social networks we wouldn't have the rights to send unsolicited messages to their chosen recipient.
The exceptions to this rule are Telegram and Twitter, where we have a bot which can be referenced in a reply to a message and the recipient will receive an automated notification.