How do I tell MailCOPA to leave my messages on the server so I can also collect them elsewhere?

Go to the MailCOPA email program Configuration menu and select User/Account Details.

MailCOPA Email Program Configuration Menu

Click your mail server entry under the Incoming Mail Server branch.

Now you will see several options including "Leave Mail On Server" and "Leave Mail On Server for X Days"

MailCOPA Email Program Mail Server Configuration