To some extent this is already something anyone interested can take a look at - for a browser to infringe on privacy, it would need to either send user data to a location, or disable security/privacy features on some subset of websites.
Open source apps like Lulu allow you to inspect details of network traffic, and tools like mitmproxy let you intercept and break the SSL on them to see exactly what is transmitted. Looking for traffic to/from tracking or other services would indicate an issue with the latter kind of issue, and can again be checked through those kinds of tools.
Are there any other ways a browser could infringe privacy? (I’m ignoring things like snooping around the disk for data, which is easily detected using open source apps, and traffic monitoring would reveal) - this should make it possible for as many people as possible to verify the privacy properties are as expected. To help make analysis easier, you could compare the differences in traffic between Orion and other browsers, and focus efforts on any differences - the difference is likely to only be the update check (if enabled) and bookmark icon fetching.