To give access to your local SQL database to an external user in SQL Server Management Studio (SSMS), you'll need to follow these steps:

Create a Login: First, create a login for the external user.

In SSMS, expand the Security folder, right-click on Logins, and select New Login.

Enter the login name, select SQL Server authentication, and set a password.

Create a User: Next, create a user for the login in the specific database.

Expand the Databases folder, select your database, expand the Security folder, right-click on Users, and select New User.

Enter the user name and link it to the login created earlier.

Grant Permissions: Finally, grant the necessary permissions to the user.

Right-click on the user, select Properties, go to the Securable tab, and click Search to add database objects.

Select the objects (tables, views, stored procedures, etc.) and grant the appropriate permissions (e.g., SELECT, INSERT, UPDATE, DELETE).