Now I think there may be several different issues with comments these days and I wrote a short post not long ago about how to potentially resolve some of them so if you find the info below doesn't help your particular issue then you can read that earlier post by clicking here.
So here's what we found...
The issue ~ Every time my friend tried wanted to leave a comment she was prompted to sign-in, however upon signing in, she was returned to the comment area where she was prompted once again to sign-in ~ and so on, in a vicious loop. Terribly frustrating.
The fix ~ We realized that if she went directly to her own blog and signed in there, you know, through the top bar on the right hand side, that she was then "officially" (for lack of a better term) signed into Blogger and could then go to any blog and leave a comment.
This is something I automatically do every day without thinking about and if you build it into your own daily online routine it may help you overcome this issue as well ~ here's hoping!
Comment buttons above can be found by visiting the Elements/Buttons section on my blog.