bbPress.org

On line 2368 of functions.bb-template under bb-includes dir of your bbPress resides the code you want to change.

bb_current_user_can( 'edit_users' )

Editing a core file is not recommended but I am unsure of how to mask it to call a custom function instead of the core function.

If its unmaskable for now (architectural) and you want a fix, then either you can change that bb_current_user_can( 'edit_users' ) with bb_is_user_logged_in() but you will have to do it again when you upgrade (or till someone posts a recommended fix) or you can append this line in your profile.php after the call to the bb_profile_data()

<?php

if ( bb_is_user_logged_in() ) {

echo $user->user_email;

}?>

[untested but should work]