To Sync SQLServer authentication logins between mirrored instances:
First create the login on the source database.
Execute the below code in ssms on source server master database
select name,sid from syslogins
Copy the name and sid and use the same sid when you are creating the login in the mirrored instance using the below code
create login with password = '*******',sid = paste the sid here with no quotes,default_database = yourdatabasename, default_language = English
The default_database = yourdatabasename is not necessary as sqlserver by default will choose master database
Once this is done you can assign permissions to the login to a specific database and the log shipping or the mirroring takes care of the user permissions in the mirrored or logshipped database.
Hope this helps.