I solved this issue through a 'hack'. Initially, I had hard-coded the web service as 'http://www.mysite.com/WebService.asmx..', but now I'm capturing the domain name on which the flash file is running using a LocalConnection and then I build the path for the web service based on the domain name returned. My code now looks like this:
var localDomainLC:LocalConneciton = new LocalConnection();
myDomainName = localDomainLC.domain();
webServicePath = "http://" + myDomainName + "/WebService.asmx..'
This way, I don't have to worry, whether the request is coming from 'http://www.mysite.com' or 'http://mysite.com'.