it's a little complex, please bear with me. You must understand how WordPress handles the various user roles: only Authors, Editors and Admins can write and consequently edit their own posts. Contributors and Subscribers cannot usually publish posts. It's explained in more detail here: http://codex.wordpress.org/Roles_and_Capabilities
P2 offers an option which lets "all users" write posts (under Appearance - Theme Options) - but the post owners are still governed by WordPress. Therefore, Subscriber and Contributor posts show up in P2, but they can't be edited by either user role.
The solution to your problem is to make every registered member an Author. That way they will be able to edit their own posts on the front page.
Hope that explains it.