If copyright holders want to take action, their complaints will go to the ISP subscriber.

So, that would either be the entity operating the public wifi, or yourself (if your mobile data plan is associated with your name).

If you're in a country where downloading copyrighted material can have legal consequences (eg, the USA and many EU countries), in my opinion doing it on public wifi can be rather anti-social: if it's a small business offering you free wifi, you risk causing them actual harm, and if it is a big business with open wifi you could be contributing to them deciding to stop having open wifi in the future.

So, use a VPN, or use wifi provided by a large entity you don't mind causing potential legal hassles for.

Note that if your name is somehow associated with your use of a wifi network, that can come back to haunt you: for example, at big hotels it is common that each customer gets a unique password; in cases like that your copyright-infringing network activity could potentially be linked to you even months or years later.

Note also that for more serious privacy threat models than copyright enforcement, your other network activities on even a completely open network can also be linked to identify you, but for the copyright case you probably don't need to worry about that (currently).