I have the same problem. When I'm logged out I can see all the games available on GoG connect, but when I logged in they all dissappear, but I have couple of them on Steam. My steam profile is public. I tried on 2 PC, and in multiple browsers, tried clearing cookies but nothing helped.

In the past there used to be a problem of the GOG Connect thing/Steam API reaching its maximum limit at some point, after which the whole thing just stopped working.

The very first time I used GOG Connect, I had to wait two days to get the games. I don't know if the same thing is still a problem or not, and whether it is causing these issues or not. I got my games this time immediately after the Connect thing was opened, so no problems for me.

But the only things you can do are:
1) wait
2) contact GOG support.

Do they inform you when a game of yours is on connect?
Wait, of course not, how would they know? :P

I'M STOOPID

Make sure your Steam profile is set to public. Mine is usually private, so when I want to use the Connect, I have to briefly enable it to be seen by everyone on the Internet. This time I got 5 games out of it, then set it right back to private.

Echoing what others have said. Contact support. Firek had me up and running in 15 minutes after submitting my ticket when I experiencing the same issue.

EDIT: because I cannot type today