Text msgs are not deleting across devices when deleted on one device

Have you set up iCloud Syncing on all of your devices?

Set up iCloud for Messages on all your devices – Apple Support (UK)

All your devices will get the same iMessages if they are signed into your Apple ID. But turning on iCloud Syncing is what keeps the changes you make (editing, deleting, etc) after you receive them coordinated.

Glad it helped!

Note that if you want to get your SMS (green) messages on all of your devices, see these instructions:

Forward SMS/MMS text messages from your iPhone to your Mac or iPad - Apple Support

If you've already done that, great!

They see what you sent. If you sent the edited version, they will see the edited version. What your recipient sees is unaffected by any settings you may have for syncing across devices. It's just like if you delete a message on any device, it doesn't delete it on the recipient's device.