Question / Problem:
When using integrated security in MSSQL and connecting and writing to the database each user needs to have permission.
When integrated security is turned on the Windows credentials are used rather than what is specified in the connection, in this case the mapping credentials. So if the only user with permission is the user specified in the connection string other users in Design Studio won't be able to write to the database.
Answer / Solution:
Add each potential user who will write to the database to have permission.