Without token you can only access public information.
Public data
From RestFB homepage :
// It's also possible to create a client that can only access
// publicly-visible data - no access token required.
FacebookClient publicOnlyFacebookClient = new DefaultFacebookClient();
If the user does not protect his posts, then you can access everything without token. But most of user do protect their data and then you need a valid user access token to read the data.
Private data
When you say "FB user already allows to access his/her info from my app" it means that the user has clicked on "Allow app" in the web browser and at that moment here Facebook will give you a token. You can after use that token with RestFB :
FacebookClient facebookClient = new DefaultFacebookClient(USER_ACCESS_TOKEN);
User user = facebookClient.fetchObject("me", User.class);
out.println("User name: " + user.getName());
By default, the token will expire a few hours later. If you ask for the offline_access
permission, the token will be valid for ever (as long as the user does not remove the permission for your app in his settings). You should store that token in your database to be able to use it when you need.
Get the user token
You cannot get the user token with RestFB. On the RestFB homepage, you can read :
Non-goals: [...] Providing a mechanism for obtaining session keys or OAuth access tokens
Because you need a browser to do so : the user has to authenticate and authorize your app on Facebook website (the popup that shows).
What you can do is to have a PHP page on which your users have to go to authorize your app. You can read this stackoverflow answer that explains how to use the Facebook PHP SDK to do so.
Hope that helps !