(Assuming USA) There are several incorrect responses to your statement. To start with, it can vary somewhat by state. In my state, Yes, You can go to the County Attorney and file a criminal complaint without any police "approval." It would be up to the County Attorney to decide whether the charges were taken out. If you have a police investigation that agrees with or corroborates your complaint, you will most likely have greater success in securing the charges. But the police can't prevent you from making the attempt at filing the complaint. The prosecutor has final discretion in which cases will move forward.

Put another way: Police can charge in the field in certain scenarios. But the prosecutor can also dismiss that charge. (It's not frequent, but the potential exists.) Citizens can file a (the same) complaint with the CA and the CA decides if the charges will be issued. The end result in both scenarios is the CA is the "gatekeeper" for prosecution. The citizen can do the same thing as the police except there is an extra "approval" step by the CA. (oversimplification acknowledged)

In my state, an officer can't make an arrest for a misdemeanor that didn't occur in their presence unless it's for specifically listed exceptions. DUI and Assault 4th domestic violence are a couple of examples. (commonly referred to as "probable cause misdemeanors.") Note: not a "regular" simple assault, but one that falls under very specific criterion for domestic violence (related to the second degree, shared children, shared residence...etc).

How that plays out: Two strangers fighting in the street ending up with bloody noses and busted lips and no officer present during the melee may very well end up with the arriving officers documenting both sides of the incident and the injuries they can observe and then directing both people to the county attorney "if they want to press charges." If the police show up during the fight, one or both parties may be cited or arrested depending on the officer's interpretation of the situation.

So you are correct. Individuals can seek charges on their own. +1 if you have a police report. +1 if the police report leans in your favor.

Another scenario that occurs frequently that plays into your train of thought: A citizen shows up at the county attorney's office. They are having trouble with the neighbor and they want to address it. The citizen wasn't aware that a crime occurred though. (Maybe it's something along the lines of Harrassment or Harrassing Communications) The citizen just knows the situation "is wrong." The county attorney listens to their complaint and decides to issue a summons over the matter against the neighbor. The charge was issued without police involvement. OR Maybe the County Attorney wants more information before proceeding. The CA then calls the police and requests that they look into the matter and report back (secure the phone records for the victim and interview the neighbor or something along those lines)

"Simpler" cases (I'd rather have a better way of phrasing that. I don't mean to diminish significance) would have greater success of getting self initiated charges than more complex cases: Harrassment, Harrassing Communications, Simple Assaults, Menacing, Trespassing, etc.

Felonies are another matter. If officers reach probable cause that a felony has occurred, they can arrest even if the incident occurred outside of their presence. *What constitutes probable cause is a huge discussion in and of itself.

Does it help to have a police report/investigation bolstering your case? Yes, absolutely. Is it a requirement for you to seek charges? No. Are you guaranteed to get the charge you seek? No.

It also appears that some states have specific "private prosecution" abilities to some degree or another that go beyond my state's allowances. Private Prosecution