Security issues? Mind letting us know which host you were with that has these security issues?
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What your asking is possible with a small modification to the Apache config which is documented on the Plesk site. Although it's a server wide feature that would allow every ftp account to see their site through www.domain.com/~username
I can understand their position as I myself won't make this modification. I prefer to give the client instructions on how to set their dial-up connection to use the server's DNS servers so they can see how their site looks on the new server. Of course this does require the client to have an account on the new server.