linked server be made using the login's current security context
From BOL:

Be made using the login's current security context

Specify that a connection will be made using the current security context of the login for logins not defined in the list. If connected to the local server using Windows Authentication, your windows credentials will be used to connect to the remote server. If connected to the local server using SQL Server Authentication, login name and password will be used to connect to the remote server. In this case a login with the exact same name and password must exist on the remote server.

http://technet.microsoft.com/en-us/library/ms188477(v=sql.105).aspx

Using the login's current security context will use whatever user is logged in to the local machine.  Be it Windows authentication or SQL authentication.

So, in your example above, if JOEsmith logs into server A and runs a query against server B, it will execute the query on server B using JOEsmith, as long as JOEsmith is a valid logon for server B.