This depends on what you have set Patient Visibility to under Settings -> General -> How do you manage your customers?:
If this is set to Visible to all users, all users regardless of user role will be able to assign a user as responsible for a patient.
All users will see a dropdown box labelled Responsible within Edit Details on the Patient tab of the Patient Summary:
If this is set to Visible to responsible user, only site administrators can assign any user to any patient via the dropdown list.
All other users will only be able to make themselves responsible for any new patients they create (or have created in the past). These users will see a checkbox in place of the dropdown list against Responsible:
Users other than Site Administrators will not be able to make themselves responsible for a patient they did not create, they will not be presented with any options under Responsible when editing patient details:
Need More Help?
We understand that not everything is black and white, so if you need some help, click "Submit A Request" ticket and one of our team will help you out as soon as possible.