One way to do it is using the checksum and cid tokens in the URL as mentioned in the other posts.
If you don't want to be limited to users needing you to send them the URL in order to access the form, there is another way using the relationship-permission extension here : https://civicrm.org/extensions/relationship-permissions-acls
The extension will create ACL permissions based on the relationship permission (the relationship permissions are usually only to access the contact's dashboard.)
You can set the first 'existing user' as the current logged in user and for every other contact, set 'existing user' to auto-fill or select list. Make sure that 'enforce permission' is enabled in the 'existing contact field'. If the logged in user has the permissions set to edit those he's related to, he will be able to select then in the webform and will not have access to any other contact.