Beyond the information Apple provides to developers (provided you opt-in), Chirp uses no third-party analytics frameworks. Chirp does not log personal information about how you interact with Twitter.com. Chirp does collect some strictly anonymous usage data, such as how often people post a tweet, or how often people load their timeline. You can opt out of this in Chirp settings on your Apple Watch. In TestFlight beta builds, Chirp may log crash reports to your device that you can explicitly choose to share (it will not leave your device otherwise). These reports do not contain any identifiable information. Simply put, your data is your data, and it will not be shared in any form to any third-parties.
Tl;dr Chirp collects completely anonymous usage data that you can opt-out of.
With the addition of the username color feature, your unique User ID is securely stored on a Nodehost Server, along with your selected color. No other information about your account is stored. If you wish to remove this data from the server, selecting a white username color in the app will have this result. As mentioned above, Chirp does collect some strictly anonymous usage data in order for me to better prioritise my work. If you're wondering just how anonymous that data is, take a look for yourself, here is an honest example of my database:
Tl;dr If you choose not to use the username coloring feature, your User ID will not be stored. Some anonymous data is collected, but you can opt-out of it.
When signing into Chirp, it requests specific permissions from your Twitter account, such as the ability to post a tweet. This is very simply because in order for Chirp to do anything — something as simple as posting a tweet or following someone — Twitter (rightly) requests that you give the client the ability to do these things. Chirp never performs any actions without you clearly requesting it to (such as by tapping a button). This is very simply required in order for the app to function whatsoever.
Tl;dr Chirp requests access to your Twitter account, but no actions are taken without you triggering it.
Not only do I not care about how you use Chirp (aside from if you want a new feature or find a bug), but I simply wouldn't have the resources to store your data even if I wanted to.