This might makes some people angry, but here's in my two cents.
When folks talk about armed security at our schools, it usually means hiring a contract guard service. Do not get me wrong here, there are MANY MANY good folks in the contract guard biz. However, most that I have dealt with in over two decades have been little more than warm bodies in a cheap uniform provided by a guard contractor who is really little more than a personnel agency.
I have only met a very small handful of the guards I have worked with over the years that I would trust with a sidearm. The biz is set to a price, not a standard. Having this kind of service may be cost effective for the district...but in the addage of "you get what you pay for" opens up a whole can of worms on other problems.
When I was growing up, there was no such thing as school police. In my district, the Principal was the highest authority for everyone on campus. If things got beyond that, then the Sheriff's office or HPD was called in. Now, just about every school district of any size has its own police force or contracts with the local Constable's office for such things. Most of the schools I have seen these days have a school district cop stationed on site during the day. Some of these campuses are way to big to be properly serviced by one officer.
The real answers will have to come from each individual school district in each area. However, my firm recommendation would be for them to steer clear of contract security guards or "volunteers" in favor of someone with full police powers, training, and authority.
I have known many teachers who were proficient and knowledgeable with firearms. But having them armed on campus is also a can of worms.