If you care about your privacy you shouldn't use any Google's product, no matter how good they are. It's how the company make its income, using your privacy.... Who knows if in 10 years we learn that in fact, it was another shady product
This is almost true for anything which isn't open source.
I have read Authy is far better than Google Authenticator.
I would recommend staying away from Authy, but if you are to use it then make sure to turn off multi device in the settings. This prevents recovery from other phones. If you ever want to transfer to another phone though this option will need to be enabled.