Well, I feel dumb, but the good news is chess.com has an option to turn this notification off. Under settings, then under alerts you can turn the friend log on/off notification off. I looked before, but missed it.
Friend Sign ON/OFF Notification

Because I didn't know where the setting was. For others that might have the same problem the live chess chat setting is within the live chess display. The settings button is in the lower left. Here is a screenshot: http://prnt.sc/el7kw1

The OP was asking for something else and based on your first comment I thought you were too. That setting, for chat, is also available where I mentioned.

Martin you are of course right. I was originally referring to sign off/on notification also, but I guess I got confused.

it comes up bottom left hand corner for me problem is the time is there wenn zoomed in but i prefer to see the notation

Is there also a way to disable this feature the other way around — so that my friends don't get alerts every time I log on? Seems like a privacy issue that we should be able to set a boundary on as users.

How do I turn them on? 2021 edition.
The site recently had a chance where going to Live Chess settings actually loads within Live. You want the Alerts tab for those options.

I can't find the setting. In the Alert tab in Live Chess settings, the options are Challenge Alerts, Event Announcements, Game Chat; no mentioning of notifications.

Working in 2021: https://www.chess.com/settings/live
I changed "Friends Alerts" to "Requests Only". Remember to click "Save" afterwards wink.png
Good luck!
Does it bother anyone else that the friend notification occurs directly on top of your time during a game? I know you can click these to make them go away, but why on Earth would it be put there!? I'm contemplating removing all of my friends so that this will not happen! Please help us chess.com!!