How do I stop my iPhone messages from appearing on my Mac? We have multiple iPhone and Mac products, but I need to correct one specific problem.  I only use Messages on my iPhone; but for some reason when I am near one of our Macs with my iPhone, my notification sounds for messages incoming to my phone do not play and the messages are appearing on my Mac.  If I delete the messages app from my Mac, will it screw things up on my messages in general?
No. Opting out of message delivery on one device (Mac or iPad) doesn't change delivery to the iPhone. 


Go ahead and disable handoff if you don't want that enabled. 
go ahead and sign out of messages on the Mac if you don't want that enabled.