This is called a qui tam action. It's a concept that's been around a long time in the English legal system (predating the US), to encourage people to help the government enforce its laws. The more contemporary system is to give private parties the right to sue on their own behalf for some wrong (known as a "private attorney general" system), which is how things like civil rights claims tend to work; the difference with qui tam is that the government is the plaintiff (source)
It's possible when the law says that a claim under some section may be filed by a private person on behalf of the government; at the time the article was written, the false marking statute (35 USC 292) stated that "Any person may sue for the penalty, in which event one-half shall go to the person suing and the other to the use of the United States." In a qui tam action, the government is the plaintiff; it's just that instead of a public attorney handling the case, it's handled by a private party.
There were three active qui tam statutes in US federal law as of 2009 (source: CRS): the False Claims Act (31 USC 3730), the patent marking claim, and an Indian protection statute (25 USC 201). The patent marking qui tam provision has been repealed, leaving just the two. For the Indian protection statute, I can't find any reference to any special procedure; for the False Claims Act, there is a special procedure included in the law.
In the False Claims Act, when an action is submitted, the government is notified and the complaint kept sealed for 60 days while the US government decides what to do. If the government wants to take over the case itself, it can do that; if it wants the private party to run the case, it can do that too; if it wants to dismiss or settle the case it can do that as well. It can also ask for the time to be extended, or for discovery to be delayed to gather evidence or prosecute a separate criminal or civil case. If the government lets the private party run the case, they can still intervene later if they have a good reason.
After the case finishes, the private party gets 15-25% of the judgment or settlement (if the government took over the case) or 25-30% (if the private party ran the case), plus attorney fees. If the facts the case was brought under were publicly known when it was filed, the amount shrinks to 0-10% if the person supplied information to the government that wasn't publicly known when it was supplied, and the case is dismissed if the person just learned about it from public sources.
For the false marking statute, a district court actually found it unconstitutional in part because of the lack of the extensive procedure found in the False Claims Act. There, the procedure was just to file the case like normal; the clerk had to tell the Patent Office that a claim had been filed within a month, but one issue was that a settlement could have happened in that time and bound the government. It's possible higher courts would have ruled had that section not been repealed later that year.
As for other countries, the idea came to the US from England, but the Common Informers Act 1951 eliminated it there. However, private prosecutions (which do exist in England) are similar in that the case is on behalf of the Crown, but handled by a private party; they're different in that the private party doesn't get any part of the judgment. While it's hard to prove a negative, I can't find any evidence of qui tam provisions outside the US.