I'm worried they are going to access my texts. How I disable this?
You sure she wouldn't? I have done it for my kids.
If your mother hasn't she can call in and make changes to remove it, or log in online with her account to make changes.
Are you there account owner?
No I am not. I am on a family plan--would one of my family members signing up cause this?
Immediately prior to being informed about my subscription to integrated messaging, I got a message about my account being accessed from an unfamiliar location. Immediately after, I received a number of Google verification code texts, so I am very worried my accounts are being compromised because now my texts are.
I've never installed Verizon Messages on my phone, so will my messages even be synced?
The account owner can do what they want with the lines they are accountable for. If you don't want them to have access and control like that you have to start your own plan or move to another carrier.
My worry is that the account owner, my mother, would be very unlikely to sign me up for such a service. I'm worried someone outside of my family has enabled this service.