Messages App Unable to send in Ventura

All users including yourself need to sign-in to iCloud and Enable Messages in iCloud on the Macs. Via the Messages App and the Settings screen (CMD+Comma).

All users need to do a similar thing on their iPhones. Settings -> Name at the top -> iCloud -> Show All under Apps using iCloud. Make sure Messages is turned ON and when you tap it make sure Sync this iPhone is turned on.

All users on iPhone need to go to Settings -> Messages make sure Messages is turned ON, edit the Send & Receive addresses for the email addresses and phone numbers to be selected. Then check Text Message Forwarding and select the Mac's / iPads you want to forward Messages to from the iPhone. Make sure Send SMS is turned on so you can text non-Apple phones using the cell carrier. Make sure MMS is turned on as well as Group Messaging. The remaining options are optional nice to have settings.

Individual users control their own notifications. If they wish to disable notifications they need to each do that individually. There are options to mute a group chat or an individual chat.

When sending texts via Messages if the sender and the recipient are on iPhones the messages are normally sent and displayed in blue bubbles. That indicates the message was sent via iCloud. When the texts are green, that shows them message was sent via the telephone carriers SMS / MMS technology. Sometimes there's a glitch and the iPhone will send a message that is green. Non-iPhone users do not have Messages and they do not have iCloud so all messages to Android will be in green. There are limitations on the cellular text data network that do not occur on iCloud.

Blue Messages are typically far more secure than Green messages. The Telco companies log all SMS/MMS messages so stating their privacy is greater is a fallacy. iPhone Blue messages are encrypted end-to-end, in transit and at rest, Carrier SMS/MMS messages are not.