What would be the risks for running a public node, are there any risks at all?
Unless it's illegal to use bitcoin in your country, I'd say close to zero. If you route everything through Tor (and therefore obscuring your activity to your Internet provider as well) even less risk. I honestly can't think of one risk in that case.
Can I also use the public node myself,
or should i rather use a private node for my private wallet?
There's no such thing as a "private node". You're either a node of the network or you aren't. What you meant is that you can use your locally accessible wallet software which will connect to your public node in a private manner, and yes you can do it. You can install a Bitcoin client (such as Bitcoin Core) which will download you the blockchain, and you can setup an electrum server, which will connect with your Bitcoin node and your Electrum lightweight wallet software, to serve you your balance privately.