I have been invited to a new private Google+ community and the "accept invitation" button just does not work (Firefox 27), also "Loading..." label is displayed if I try Chrome (32). Restarting the browsers and disabling all the extensions does not help. Some of my mates invited to the same community report the same issue. Other communities and invitations seem to work just fine.
A mate of mine (not present on this site) has provided a working solution, let me share it for those who might be having the same problem:
Click on the Home button, click "Communities" and you'll see something that says "Community Invitations". Click on the community below, and click " Accept Invitation".