How can I keep my iMessage on iPhone, but stop them coming through on my Mac? I don't want my private messages popping up on my Mac, so is there a way I can remove my account on my Mac, but keep it on my iPhone?
Open Messages and open the preferences.

Select the Apple ID you want to delete and then sign out. 

No more messages should arrive from that point on. You could reboot if you want to be sure things are reset.