When running a local node, the most other people could possibly see is that “x IP is running a Monero node”

When connecting to a remote node, the node can see:

• Your IP address
• When you submit a transaction (which could link your IP to your transactions)
• The last block your wallet synced (which could be used to determine when you usually use/spent monero last)

It’s also possible for a remote node to feed your wallet a manipulated list of decoys, which can reduce the anonymity of the transaction you submit by allowing the remote node to simply remove the fake decoys to find which isn’t the decoy (you.)