Answers
Do messaging apps read your messages?
Some can and some cannot. End-to-end encrypted services — Signal, WhatsApp — cannot. Services that screen, scan for malware, or offer server-side search can, and generally say so somewhere in their policy.
The honest way to answer this for any given service is to look for what it does that would require reading. Server-side search, link previews generated centrally, spam filtering and content moderation all require it.
A service that offers those and also claims end-to-end encryption is doing the work on your device, or not doing it.
Naven reads messages in order to screen them and says so plainly, because the screening is the product rather than an add-on.