To my understanding, that was the original intent in both British and American jurisprudence. A Citizen would present his complaint and evidence directly, and if the GJ agreed they forwarded a Bill of Particulars to the District Attorney. That the DA's and Courts have usurped that provision is one of the reasons victims receive little justice in our current system.
Make it so the Complainant has to foot the costs if the GJ issues a "No Bill" and will cut out the frivolous suits, both criminal and civil. 'Loser Pays' is a good idea that should be implemented across the board.
I can see some potential problems in that, but a more modern version has some possibilities.
To start with, a private grand jury would have 15 people employed by it. A retired judge or other legally competent arbitrator; a attorney who is experienced with criminal prosecution; and a third competent arbitrator who just observes the entire process, not contributing to it, but is the third vote in a panel of the three.
The remainder are 12 educated and objective jurors. And this is important. These are not “peers”. They hold college degrees as experts in different fields, so are competent to examine and analyze evidence. In effect they are expert “witnesses”, when examining the evidence.
Ideally, the jury should have a corporate law attorney, statistician, physicist, an information technology and computer expert, an electronics technician or engineer, a linguist, a criminal forensic expert, a pharmacist, an ER nurse, a gun and explosives expert, a chemist, a historian, and several research assistants for the others.
Of course, there would likely be overlap in expertise, as well as other experts needed on an intermittent basis.
This would be prohibitively expensive for ordinary criminal cases, and there is not a big backlog of cases presented to grand juries currently. So this private grand jury would mostly work to examine extremely complex white collar crimes, and crimes of the wealthy against the wealthy.