DMCA only come from trackers that are open to the world. ISPs don't snitch on you. They only do what is required of them by law, usually. (Forwarding DMCA complaints).
The copyright companies, connect to these trackers and perform regular torrent client requests, such as "hey, who is seeding this torrent". And the tracker responds. They take note of the IPs and do whatever they do.

Deep Packet Inspection (DPI) can only find out that you are doing a P2P connection and possibly torrenting. Which in of itself is not illegal.

With private torrents you are blocked from the DMCA copyright companies because they cannot find out what torrents are on the trackers because you need the metadata of a torrent & the password to get the peer list.

A VPN only solves the problem where the ISP is hostile to you, the consumer and obfuscates who you are connecting too. With DPI and packet analysis (which is slightly expensive) they could figure out you are torrenting via a VPN with a high degree of certainty. Butt at the end of the day, all they would see is encrypted packets. This data is no different than telling your torrent client to Force Encryption, which everyone should do and I get annoyed all the time when people don't have it on.

Tldr,

• VPN only blocks ISP from seeing unencrypted P2P traffic and makes it harder to identify.
• DMCA companies can only access public trackers peer lists where it gets its information from