Windows Live Mail is an email client which (like Outlook, Outlook Express and Hotmail) downloads emails to your computer so that they can be read locally. The default setting would be to atomatically delete those emails for your email provider's server, so that they wouldn't be available via any other machine. However if you've got the same emails on two computers you must have over-ridden that setting and you've now got two totally separate copies of each email. Deleting one won't affect the other.
If you want to check that your emails are stored locally (and not on a remote server) simply disconnect your machines from the internet. If your emails are still available then, clearly, they're stored as files on each computer and you can safely delete them on one machine without affecting the other. (It's no different to deleting a picture on one of your computers, that you've also got on the other. The second copy won't be affected in any way).