You have to stop sharing the same Apple ID with each other. The messages, among other content, are also being shared. Use separate Apple IDs and look into 👉 Family Sharing.
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It is just not Messages, but Contacts, Notes, Calendar Events etc. will also be shared between the devices that use the same Apple ID (iCloud account). Your best option is to use separate Apple IDs.
If you want to continue to use the same Apple IDs for two devices but are only concerned about Messages then there is a workaround which is not clean but will somewhat do the job.
For the Messages isolation - Take one of the iPhones and open Settings ➔ Apple ID (Your Name at Top) ➔ iCloud ➔ Messages: Set this to OFF.  Then, open the Messages app. Delete ALL Message threads (Yes I told you, it is not a clean method). Then, open Settings ➔ Messages ➔ Send & Receive: Ensure that ONLY ONE Phone number is checked and that is the phone number of that particular iPhone. No email accounts should be checked. On the same screen, select start new messages from and choose the number of that iPhone. On the same screen, select Text Message Forwarding and set the switch to the OTHER iPhone to OFF.
The above will address the two concerns that you had mentioned in your post. It is not clean and it does not really separate the two iPhones completely, however, it may be just what you need. I still recommend that you use separate Apple IDs for both iPhones if you want to completely keep the data separate.
Axel F.
