Running a Lighting Rod
Running a Lightning Rod does not affect your privacy any more than running a public Bitcoin node. The only way your privacy can be compromised is if you are tieing your Bitcoin node / Lightning Rod to your identity.
Connecting to a Lightning Rod
Connecting to a Lightning Rod with your lighting node can compromise your privacy, the owner of your Lightning Rod could see and track every command you are running and the data being sent with those commands. It is recommended that you do not connect to a Lightning Rod as your primary way of accessing the Lightning Network for this reason.
Running a Lightning Rod does not leave your Bitcoin node open to attacks, your Lightning Rod will not pass any RPC command to your Bitcoin node that would affect your node or wallet. You can view the list of commands your Lightning Rod will allow, you can visit here. If you wish to add or remove any commands you can whitelist or blacklist the commands. Examples on how to do this can be found in the user guide.