How can you create on-screen text messages using video editing programs?
These are actually really simple shape/text overlays with a bit of animation. You could probably do a quick and dirty version (though, honestly, it wouldn't even look that dirty!) right in your editor by dropping on some shapes and writing over them in text.

Alternatively, you can create high-resolution versions of the messages in something like Photoshop or Illustrator (or GIMP, if they aren't available) and export them as PNGs which you'll then drop into your project (I use Premiere Pro so I don't know how well FCP talks to Adobe programs - sorry!).

Personally, I use After Effects for things like this and, again, you can either create the text messages in Ae or use Ps/Ai to create the individual layers then import that into your project.

In terms of making it work, there are multiple ways of doing this, too. You could just create a single image and mask areas off until you need to see them (at which point you can animate the mask to reveal the messages or just remove it for simplicity). Or you could animate each message coming in with a swipe or a pop or, perhaps, you could have the sender scale on, the message bubble pop on and the message write on one at a time - entirely up to you!

One more thing - don't worry about sticking too close to a particular design, or about copying the little flick at the bottom of the bubble - as long as you can tell at a glance (ie: without reading the sender info) which messages are incoming and which are outgoing (different colours/left and right margins) you'll be fine ;)