aiContactSafe doesn't require you to give full access ( 777 ) any of it's files but you need to set that way the temporary folder ( at least ) in order for Joomla to work if the server is not properly setup.

This is a hosting problem and professional hosting providers are setting their servers so the php is running as the user that you would normally use to access the server by FTP.
So if you are having problems with the access rights you can either ask your hosting provider to setup on his server suExec or something similar ( he should know what is this ) or activate your FTP layer in Joomla's Global Configuration ( see the instructions on Joomla's documentation about this ).