I see this situation over half the time I visit someone's profile. But most of the time it has nothing to do with G1, but the way I have my various computers setup (see below):
For example, here's a screen shot of RG's profile page:
Notice the circled area where the website thumbnail should appear, but does not.
However, 90% of the time its not G1's fault, but rather how the firewall is set up on the person's computer. If the firewall is blocking Private headers from being sent then you will often get the result shown above. So I'd recommend testing by visiting one's profile with the firewall completely off and that will confirm whether it is a firewall setting.
Also keep in mind that firewall settings on the person's computer can affect many different aspects of G1. Usually you can fine tune each site in the better Firewalls to allow and disallow just what is needed to make things work the way you want. Sometimes its a balancing act.