All types of verifications should have been done; background, criminal, credit and job check.Your agent should have known all this and should have provided and you should have received the reports and made final call. 

I used to have a neighbor that somehow got my number and she would call over the silliest complaints.

I have an applicant that submitted and have run the credit, criminal and rental history.  

This includes who they can lease to (600 score, 2.5x rent income, no evictions, no criminals).Lastly, the threshold for being a PM is low, so most are bad at it... like you, they all are in it for their own business and income, not yours.

With a voicemail at the end so that if someone does have a complaint the broker will get it.

Should I take steps to have a prosecutor file criminal charges of trespassing and vandalism?

At DORA you can look up your HOA and make sure they have registered (you can also file a complaint) and at the Secretary of State website you can make sure they have been filing required documents as per requirements.

No complaints. 

Are these common criminals brokering without a license or licensees?

Filing a complaint with them is free and they do try to look into all the complaints (even if it takes a while to start).