How can I see my child's text messages without them seeing mine?

I can think of one way. You need to know your child's Apple ID and password. And you need to assure that the phone is charged overnight, every night, and that iCloud Backup is turned on. This will create a backup when the phone is locked and connected to Wi-Fi or power. Alternatively, any time you want to view messages just back up the phone to iTunes on a computer. Then get a backup extractor program such as https://www.reincubate.com/labs/iphone-backup-extractor-how-extract-files-iphone -backup-windows/ (pricy: $70/year to view iCloud backups, but the only one that works with iCloud backups).

You an also get an iPhone data extractor, such as TouchCopy and download the messages to a computer.

These methods are all after the fact; there is no way to monitor them in real time.

Other than requiring that they hand over their unlocked phone to you upon request, no, not really.

There are some programs that claim to be able to do this, however, they generally involve jailbreaking the phone which is not a good idea as it then becomes vulnerable to malware. Other service involve trusting them with your (or your kid's) Apple ID. Also not the safest thing to do.

See this article for the options that Apple offers:

How to use parental controls on your child's iPhone, iPad, and iPod touch - Apple Support

That's a rather clever workaround.

And I just thought of it as I was answering 😉

The Reincubate app has another use that I won't post in a public forum, but that solves another common request. 😉