Yes! Once you have connected a contact record to a Joomla! user, when they are logged in they will have permission to edit their own information. Note that the administrator can specify some contact information to be unavailable to the users for editing. In the 'Editing & Registrations' panel of configuration, look for the 'USer Editing Permissions' section.

And since we're talking about it - you can allow users to register on your site through Grumpy Contacts. You select the type of registration to use in Grumpy Contacts configuration - user activation, administrator approval, or no user creation at all. Note that if you do not allow users to edit their categories, you MUST set a default category for registration, otherwise no one will ever be able to register.