Well, you asked it on 2011 and Google Groups changed recently some features...

But if you want to enable replies from all users on the new Google Groups interface (hey, I use it in Portuguese, so some words here would be synonyms of real options):

1. Enter the group and choose Manage

2. On Details > E-mail Options, change Post Answers to All Members, and press Save

3. On Details > New Members, check the checkbox New Users can Post, and press Save

4. On Permissions > Basic, on Post select All Members, and press Save

5. On Permissions > Post, on Post select All Members (if you have done 4th step, probably this one would be already done too, but once, when I was creating dozen of new groups, options from these steps, 4 and 5, were different, perhaps a momentary bug), and press Save

6. On Info > General, on Post Options, check the option to Allow E-Mail Posting

7. And finally, on Members > All Members, check if someone is not allowed to post (on Posting column). If so, select them and on Actions > Post Permissions select the default option.

You seem to be thinking that a "Google account" and a "Gmail address" are the same thing. This is not true; you can sign up with any email address. However, you cannot get away from the account requirement – for obvious security reasons.