When filing a counter notification, you must consent to jurisdiction of the US federal district court.
If you're based in the US, then you would be consenting to the US federal District Court where you live. Because U.S. citizens and businesses can always be sued in the district where they live, the consent-to-jurisdiction-requirement doesn't have an impact for U.S. citizens and businesses.
However, if you're a foreign owned and operated website, you must consent to jurisdiction either where Google is located (Santa Clara, CA) or where the original complainer is located. Consenting to jurisdiction can increase your risk of being sued. It is more difficult to serve people in foreign countries with a lawsuit, unless they consent to jurisdiction in the US. Thus, before a foreign owned and operated site issues a put-back notice, it should seriously consider the risk of suit.
If the original complainer doesn't really have a good case and he was abusing the DMCA for censorship or business reasons, then your risk of getting sued was probably slim in the first place. Thus, consenting to jurisdiction wouldn't increase your risk that much. However, if the original complainer is very serious and thinks it is worth filing a lawsuit against you, then your foreign residence may be helping to protect you from a U.S. lawsuit. In this case, you wouldn't want to consent to U.S. jurisdiction. It may be better to start a new website with new, non-infringing content.
Thus, before a foreign entity sends a put-back notice, it is very important to step back and make sure that you're not at serious risk of a lawsuit or that you don't mind potentially defending against a lawsuit in the U.S.