When using AirDroid do your text messages and/or data get transmitted through AirDroid's servers or is that data kept locally between your Android device and your computer?
AirDroid for Is AirDroid Secure? has noted:
[In] Remote Connection Mode, your account credential, SMS, contacts, call logs and many other sensitive private information are all securely transmitted via HTTPS and we never store them (except your AirDroid account information, the latest location of your Android device if the Find Phone feature is activated) without your explicit consent.
For Lite mode and LAN Connection mode,
[All] data is transferred on the same local area network and not exposed to the Internet[.]
(If needed, see What's LAN Connection Mode and Remote Connection Mode?)
In Lite and LAN Connection mode, you're not sending anything to AirDroid's server(s).
My understanding is that your computer and Android device need to be on the same wireless network, yet AirDroid requires a login to their servers.
As per AirDroid's help center, you can use AirDroid without logging in or having an account. See How to Use AirDroid Web Without an Account?
- use Lite mode: find the IP address of the Android device having AirDroid running on it and use that in a web browser.
- scan the QR code : go to web.airdroid.com and scan the QR code there using your AirDroid app.
- use tethering
The first and last method can work without the need of Internet access, thereby, comes under local transfer.
See What is local transfer and remote transfer?
- Local transfer: When both PC and Android device are under same network.
- Remote transfer: When the device and PC are on different networks.
There is a catch however. If you're certain that the latest AirDroid app requires log in for its usage, then either the help center has not been updated accordingly or the log in itself isn't mandatory yet you've somehow ended up believing that it is needed.