I use Pocket Casts [1]. Can't say I've investigated privacy or other concerns but it is free and seems quite functional with a good UI.

[1] https://pocketcasts.com/

I wish there was an open source backend service for syncing. Since the clients are open source, should be technically possible to replicate the functionality of the cloud service. But maybe that's harder than just creating from scratch? Too bad though because pocket casts is definitely one of the best android clients around.