The watchtower stores fixed-size, encrypted blobs and is only able to decrypt and publish the justice transaction after the offending party has broadcast a revoked commitment state.

Source: https://github.com/lightningnetwork/lnd/blob/master/docs/watchtower.md

Here is a nice talk about WT by Sergi Delgado at BH2022. There's also a talk at Chaincode about LN privacy and WT by Sergi, but can't find it right now.

Not really. But if you are already a public node, you are already exposed to public, right? Any LN explorer can show all your channels.
A WT is only keeping updated the information about the status of your channels (last commit etc), but not knowing all your detailed txs.
Also you can have anytime a "friendly" WT peer, for example a friend that is also running a public node and you can watch reciprocally, in private.