Hi rviera71,
Thanks for reaching out in Apple Support Communities. I understand that when you're working on your Mac, your iPhone doesn't sound an audible alert when a new message is received. I can see why you'd reach out for more information about this; I'd like to help. It sounds like you're using Text Message Forwarding on your iPhone which could be causing the behavior; one of the benefits of using this feature is receiving notifications on the device you're using at the time.
If this is a feature you'd like to continue to use, but would prefer the alerts to come to your iPhone, you can use these steps to disable notifications for Messages: Use notifications on your Mac
Alternatively if you'd like to continue seeing alerts on both devices for Messages, you should see an alert on your iPhone shortly after the alert on your Mac if you haven't acknowledged the notification on the Mac.
If you'd like to disable forwarding altogether, this resource has steps to help: How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac
Best Regards.