Wandb only logs what you tell it to log. The data hosted on their server is opt-in, meaning you explicitly log what you want when you use their API.

If you don't trust any third party with your process, the local setup seems pretty straightforward (docker container), info at https://docs.wandb.ai/guides/hosting/basic-setup.

I did use some other experiment tracking software and found wandb to be easier and full of useful features. Granted I didn't try MLFlow, but wandb had almost everything I needed so no point in trying all the options. (I wanted a mix of collaborative Tensorboard+Git+DVC+Pachyderm but on windows)