Assuming that the VPN service provides logs to law enforcement, then yes. Internet companies in the United States are required to adhere to U.S. data laws. That includes VPN services. That includes the providers used by private internet access.
"Wait, what? But Mark Buffalo, Private Internet Access claims to not keep logs!" Yeah, that doesn't matter. PIA doesn't own the servers you connect to. These servers are leased by hosting providers all around the world. They are the ones who have lots of logs, even if PIA doesn't. Suddenly, hiding behind a VPN seems a lot less secure, huh?
If you're from Russia, and you use a VPN service which obeys U.S. jurisdiction, and you use it to commit fraud (shame on you!) on Russian Territory, then the FBI will likely help the Russian police bust you, and anyone else involved.
Why? There is already precedent for this:
- Nine People Charged in Largest Known Computer Hacking and Securities Fraud Scheme
- After a dozen raids around the world, police are detaining the hackers behind 'RAT' software, which lets attackers take over Facebook accounts and infect computers
- International Cooperation Disrupts Multi-Country Cyber Theft Ring
Search the internet. You'll find lots of evidence of the FBI helping foreign governments bust criminals.
Additionally, you may want to read the Stored Communications Act. If any data is logged, companies are required to hand that over to authorities. Most companies would be more than willing to help law enforcement, as they can charge them access fees.
Now what's the best way to avoid the authorities? It's easy, really.