I think there's 2 potential issues with earbuds, the Bluetooth connection itself and/or a pairing app for the buds.
As for the Bluetooth connection, I've never read about security concerns. I don't think recent versions of Bluetooth are at risk of leaking data. They might be somewhat leaky in metadata (which devices are sending signals at all).
The pairing app poses imo a larger risk. These apps are usually proprietary and might be trying hard to get information. That's a risk with any app that requests many permissions, though. A simple solution would be to not use the pairing app. Or you might look into compatibility of your device with Gadgetbridge, which is open source. https://gadgetbridge.org/