How does one prevent another iPhone from seeing Messages sent to me only?

Sign out of Messages on the other 2 phones and sign back in using the Apple ID of the person using the phone. It sounds like you are using the same Apple ID on all three phones.

Hi,

I use separate IDs for iMessages and FaceTime compared to iTunes and iCloud.

I tried adding one Apple ID to one iPhone and another ID to another when I had two iPhones.

This is fine until the computer wants to link them via the Mac's version.

You think you have them separate but the Mac links them all.

I have an Apple ID now that shows up wanting to be added on my iPad, remaining iPhone and Mac (and does even if you say No - although not active).

I removed the other iPhone from iMessages before shutting down the app.

see https://selfsolve.apple.com/deregister-imessage

For extra measure I would remove com.apple.imservice.iMessage.plist in Mountain Lion

I say Mountain Lion as this is where your Thread is.

Later versions have an IDS .plist with madrid in the name which you should remove instead.