If the user in DOMAIN2 needs access to folders/files on your server (in DOMAIN1) and you cannot set up a trust between the 2 domains, then you are gonna have to build an account in your domain, then send the username/password information to the user.
Tell the user to map a drive to the shared folder on your system, but to use the Connect As option in the Map Network Drive window. On the line, have the user use the DOMAIN\USERNAME syntax.
So, if the username you make is REMOTEUSER, then the user will put on the Connect As line:
DOMAIN1\REMOTEUSER
When the user clicks OK, they will be prompted to type in the password. Tell the user to do so.
Also make sure the user leaves the checkbox Reconnect at Login checked for next time.
Then, every time the user logs into their system (authenticating to DOMAIN2) they will get the password prompt box to connect to the shared folder on your server in DOMAIN1. The user will type in the password you gave him/her.
That is how it works when you can't set up a trust. Without a trust, you cannot access DOMAIN2's user list to be able to add the user to the security on your shared folder.
hope this helps
