Apple's press release seemed to say that much of the functionality will be pushed in the coming months via iOS updates (since the iPhone to iPhone communication isn't exposed normally except for Find my iPhone). It's unlikely that you'll need to install an app except to say you are infected.

Google could do a similar thing via a Google Play Services update (or, you know, use this as the kick-in-the-pants to get manufacturers to start updating Android to protect the public from COVID-19).