given that your web server is reached throug local.www.mysite.com and that your lightstreamer server is reached through local.lightstreamer.mysite.com, you can't use local.mysite.com as domain as it doesn't match neither host.
Hosts and domain must respect the "same origin policy".
So you should use as domain only mysite.com.
If you prefer/need a longer domain, another possibility is the following:
Web Server: www.local.mysite.com
Lightstreamer Server: lightstreamer.local.mysite.com
Domain: local.mysite.com
Hope that helps.
Note that Opera poses further limits to this policy. Check these thread for details: http://www.lightstreamer.com/vb/showthread.php?t=28 and http://www.lightstreamer.com/vb/showthread.php?t=96
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