Once the proper configuration profile has been installed from the Apple Developer Center, the new watchOS beta can be downloaded through the dedicated Apple Watch app on the iPhone by going to General --> Software update.
To install the update, the Apple Watch needs to have at least 50 percent battery, it must be placed on the charger, and it has to be in range of the iPhone.
We don't yet know what features or improvements might be included in the watchOS 4.2.2 update, as no outward-facing changes were discovered in the first two betas.
It's possible this update focuses mainly on bug fixes, and if that's the case, we won't know the extent of the changes until it sees a public release complete with release notes. Should new features be discovered in the third watchOS 4.2.2 beta, we'll update this post.