All, I think I have a solution. I did a fair bit of troubleshooting and discovered what was going on in my case. I had three calendars that I shared with my wife when we were both on MobileMe. I upgraded to iCloud/Lion first. No problems - but we couldn't see each other's calendars as expected. A few weeks later, I upgraded my wife to iCloud/Lion. Her Calendar then started having those 403 errors everyone is talking about. I tried turning the iCloud Calendar feature on and off, and even signing in and out her iCloud account in System Preferences to no avail. I even deleted the shared calendars from her system and re-added them again. Still, was getting the 403 errors. I noticed that two of the three calendars were disabled when I created a test event to then try and switch the event onto the shared calendars. Aha, I thought. I now know only two calendars have a problem. So, when I deleted the calendars no more 403 errors. But, if I re-add them, I get the 403 errors again.
The solution is this: I deleted the shared calendars from my wife's iCloud calendar list and then logged into my own iCloud account. I stopped sharing the two calendars in question and re-shared them again to my wife's iCloud e-mail address. Lo and behold, the new invitations took just fine and now, no more 403 errors. So, the source of the problem would appear to be a server-side permissions or backoffice kind of problem. The workaround is to delete the sharing properties (i.e. stop sharing) and then re-share the calendars again. I think maybe now with this information, Apple should be able to home in on a real fix but until then, try this workaround.