Re:
I think My answer will probably only confuse you more. I will speak from a California enforcement perspective, but I would guess that most states have very similar "citizen arrest" laws and requirements.
Vandalism can be a misdemeanor or a felony. It's a crime though, and people can be arrested for crimes. It may rise to the level of felony if the damage is financially high.
A citizen can make a citizen's arrest for a misdemeanor committed in their 'presence'. If you see someone committing the vandalism, or you heard the glass break and you turned around and saw John Smith there standing by it with a hammer in his hand, you can go to him and tell him he is under citizens arrest for vandalism - in some states also called malicious mischief, but vandalism is a good catch-all.
Your duty at this point is to turn Smith over to the police.
The duty of the police officer is to 'accept the prisoner' from you. In some states, or at least in California, it is a violation of law for the officer to NOT accept a prisoner. At this point the police officer has options. He/she will listen to your story and gather your information on what you saw and how you made the decision to arrest Smith
Accepting your prisoner does not mean the officer must transport the prisoner to jail, or to keep Smith in custody. If the evidence you provide is sufficient to meet legal criterial, the officer can book Smith or issue him a citation for a future court appearance.
If the officer believes that your arrest is flawed, he can release Smith right there on the spot, and just submit a report on the matter.
You may have an option later of having the facts of the report reviewed by your local DA's office. But if the officer didn't think the criteria was there, that's a bad sign as to any subsequent complaint being issued.
You should always give thought to the consequences of a false arrest. Make sure you know what you saw and that what you saw is a crime and that Smith is the one that you saw doing it. Otherwise, Smith may want to retaliate with claims of false arrest, etc.
That's the way it's 'supposed to work' in California. But everyone will have a story like yours where it didn't work. Not everything happens by the book (so to speak) and I don't know the laws of Georgia on citizens arrests. For that you may consider a casual conversation with an officer in your community, or a visit to your library reference section for the topic of the "laws of arrest, search and seizure" for Georgia.
Added note: I agree with one of the people above who thought you may have misunderstood an officer's reference to the property being privately owned, so they can't enforce laws there. That wouldn't be correct. Have them go over that part again.
If the vandalism is on-going and truly an chronic, expensive problem, you might consider speaking again with your local 'regular' patrol officer or patrol supevisor to establish something of a partnership in resolving the crimes. Some citizens are effective in communicating with their local patrolman about the problems, so when you call because you just put the grab on Smith, the officer will know as soon as he's dispatched what the history of the problem is, because he talked to you already and has advanced understanding.