How can I delete a mail on my desktop but leave it on my IMAP server so someone else can read it from their desktop?
When you use IMAP accounts if you delete mail locally, mail clients then mirror the same deletion on the remote mailbox end. That’s the nature of IMAP account mailboxes. No way round that.
So if you want certain other parties or your own other mail clients to see & keep those emails before you delete them in eM Client, then have the other partys or yourself create a custom local folder in their mail clients (outside of the account) and copy those specific emails to that folder. Once done, then you can delete those emails from your eM Client mailbox end. You might also even choose to do that yourself for any old mail before you delete it from the server.
If they are using eM Client, they can create a custom folder in “Local folders” at the bottom left (under the accounts) to keep any old server based mail.
Note: The only thing to be aware of with keeping mail locally in any mail client is you have to backup the mailer regularly Incase anything goes wrong with the computer and you then need to restore, as those messages are no longer on the remote server mailbox and only in local folders.
Short answer, you can not but try @cyberzork way and the other way is with POP.