I'll try and share my understanding. Someone more knowledgeable on the subject please feel to chime in.

Regarding criminal matters, e.g. a law was broken, law enforcement personnel are the only folks who can 1) issue citations, 2) make arrests, the two components that make up what I think most refer to as "pressing charges." Citations can take a few different forms, either a collateral citation (pay a fine), or mandatory appearance, which is requiring someone to show up and talk to a judge for their penalty. Being arrested is fairly straightforward. A person is taken into custody and formally charged, by the arresting agency, with having committed a crime.

As a private citizen, our only dealings with criminal issues would be notifying LE, and/or providing accounts, testimony, etc. Although LE folks sometimes take our preference and opinions into account in making their decisions on appropriate courses of action, but they are certainly not required to do so, and often times don't. In other words, we have no real say as to whether or not "charges are pressed."

We, the private citizens of the US, have the civil system to help us seek remedy for alleged wrong doings that harm us. These wrong doings may be criminal in nature, but don't necessarily have to be. Instead of "pressing charges" what we would be doing is seeking a civil remedy for the alleged wrong doings. In other words, trying to demonstrate in court that damages were suffered, and certain actions (or lack thereof) were the cause of said damages, and then of course request that a suitable remedy be made, financial or other.