It depends entirely on the phone and the message type, and the honest answer is that iOS and Android are not comparable here.
Full disclosure: we build Ultimate Phone Spy, a parental-control app for exactly this β web history, app activity and screen time in one dashboard. Plans start at $16/month on the 12-month option, and the current plans are on our pricing page.
Android
SMS lives in a readable system database, so a monitoring app installed with the right permissions (SMS access, or notification access as a fallback on newer Android versions) can log incoming and outgoing texts as they arrive. Google's default SMS permission policy restricts this to the default messaging app or an approved category, which is why most parental tools on Android rely on notification access or a device-owner install. In practice: expect message text, sender, timestamp and MMS metadata, delivered to a web dashboard within seconds.
iPhone
There is no API that lets an app read the Messages database on a stock iPhone. The two legitimate routes are:
- iCloud backup access. With the Apple ID, password and two-factor completion, the service reads the last backup and shows messages, including iMessage and SMS, up to the backup date. It is retrospective, not live.
- A supervised device profile. Gives limits, location and app usage; it still does not expose message contents on modern iOS.
Anyone selling "live iMessage reading with no jailbreak and no iCloud" is selling a jailbreak in disguise or nothing at all.
What does not work
- Carrier call and text logs show who and when, never content. Family plans include them; they are metadata.
- Apple Family Sharing has no message-reading feature, by design.
- "Sync your child's messages to your phone" tricks usually mean signing the child's Apple ID into your device, which breaks their iMessage and gives you their whole account.
The consent line
Reading an adult's messages without their knowledge is unlawful in most jurisdictions, and in several US states it is a criminal matter regardless of who owns the phone. On a minor child's device, monitoring is generally lawful for a parent or guardian β and the strongest setups are the ones the child knows about, because they survive being discovered.