I also had the problem of the "Share" button doing nothing, in both Chrome and FireFox, whether in the extension or on the web site.
To work around the problem I had to do the following:
1) Use FireFox in Incognito mode (also know as a "Private Window")
2) Type in the email address of the recipient instead of choosing from the drop-down, and then hitting the Enter key.
This is super-annoying and really needs to be fixed.