NeosPeeps contacts the Neos API, so by using this application you're going to also be subject to Neos' privacy policy as well.
The application stores your Neos session details as well as other configuration options locally on your PC. The data is stored in plain text so you can inspect it, and isn't sent elsewhere (with the exception of using the Neos session details to authenticate to the Neos API). Your actual password and your 2fa code are never stored outside of the running application's memory.
The application stores requested images in the system's temporary files folder to avoid re-requesting large files (like images for example) over and over again.
The application can optionally check for updates. This either happens when the user manually presses the check for updates button, or if the auto update checking feature is enabled (disabled by default for your privacy).
The update checks contact the git host API to retrieve the latest release version. Logs of such requests may be kept on the server side. These logs can contain the request path, the user agent as well as timing and network related information. The information is stored in order to keep providing the service in a secure manner and to prevent malicious actions, like (D)DOS attacks for example.