Another vote for Signal as it seems to meet your requirements. It's just like texting except your messages are encrypted. It's recommend by Edward Snowden who definitely knows what he's talking about on this subject so I'd say it's good enough for sharing data like this. There are other apps that do the same but Signal is one of the most respected and is designed to be very easy to use.
WhatsApp source code isn't available to review so you have to basically trust they do everything they say they do, and they also recently said they will be sharing some information with parent company Facebook (although not the messages themselves).
Another note, Facebook rolled out secret conversations for their messenger platform based on Signal's protocol. WhatsApp and Allo seems to use Signal as well: https://whispersystems.org/blog/facebook-messenger/
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